Question 15 Marks
Find the vertex, focus, axis, directrix and latus-rectum of the following parabolas:
$y^2 = 5x - 4y - 9.$
AnswerThe given equation is
$\text{y}^2=\text{5x}-\text{4y}-9$
$\Rightarrow\text{y}^2+\text{4y}=\text{5x}-9$
$\Rightarrow\text{y}^2+\text{4y}+4=\text{5x}-9+4$
$\Rightarrow\text{(y}+2)^2=\text{5x}-5$
$\Rightarrow\text{(y}+2)^2=5\text{(x}-1)....\text{(i)}$
Shifting the origin to the point (1, -2) without rotating the axes and denoting the new coordinates w.r.t these axes by X and Y, we have,
$\text{x}=\text{X}+1,\ \text{y}=\text{Y}-2....\text{(ii)}$
Using these relation equation (i), redus to
$\text{Y}^2=5\text{X}......\text{(iii)}$
This is of the form $\text{Y}=\text{4aX},$ on comparing, we get
$\text{4a}=5$
$\Rightarrow\text{a}=\frac{5}{4}$
Now,
Vertex: The coordinates of the vertex w.r.t new axes are $(\text{X}=0,\ \text{Y}=0)$
$\therefore \text{ x}=0+1,\ \text{y}=0-2$ [Using equation (iii)]
$\Rightarrow\text{x}=1,\ \text{y}=-2$
$\therefore$ coordinate of the vertex w.r.t new axes are $(1, -2).$
Focus: The coordinate of the focus w.r.t new axes are $\Big(\text{x}=\frac{5}{4},\text{y}=0\Big)$
$\therefore\text{x}=\frac{9}{4}+1,\ \text{y}=0-2$
$\Rightarrow\text{x}=\frac{9}{4},\ \text{y}=-2$
Axis: Equation of the axes of the parabola w.r.t new axes is
$\text{y}=0$
$\therefore\ \text{y}=0-2$
$\Rightarrow\text{y}=-2$
$\therefore$ equation of axis w.r.t old axes is $\text{y}=-2.$
Directrix: Equation of the directrix of the parabola w.r.t new axes is
$\text{x}=\frac{-5}{4}$
$\therefore\ \text{x}=\frac{-5}{4}+1$
$\Rightarrow\text{x}=\frac{-1}{4}$
$\Rightarrow\text{4x}+1=0$
$\therefore $ Equation of the directrix of the parabola w.r.t oid axes is $\text{4x}+1=0$
Latus-rectum: The length of the latus-rectum $= 4a$
$=4\times\frac{5}{4}$
$=5.$
View full question & answer→Question 25 Marks
Find the equation of the parabola whose:
Focus is (0, 0) and the directrix 2 x - y + 1 = 0
AnswerLet P(X, Y) be any point on the parabola whose focus is S (0, 0) and the directrix 2x - y -1 = 0. Draw PM perpendicular from P(X, Y) on the directrix 2x - y - 1 = 0.
Then by definition
$\text{SP}=\text{PM}$
$\Rightarrow\text{SP}^2=\text{PM}^2$
$\Rightarrow\text{(x}-0)^2+\text{(y}-0)^2=\Bigg(\frac{\text{2x}-\text{y}-1}{\sqrt{(2)^2+(-1)^2}}\Bigg)^2$
$\Rightarrow\text{x}^2+\text{y}^2=\frac{\text{(2x}-\text{y}-1)^2}{(\sqrt5)^2}$
$\Rightarrow5(\text{x}^2+\text{y}^2)=(\text{2x}-\text{y}-1)^2$
$\Rightarrow5\text{x}^2+5\text{y}^2=\text{}\text{2x}^2+(-\text{y)}^2+(-1)^2+2\times\text{2x}\times\$-\text{y)}\times2\times(-\text{y)}\times(-1)+2\times(-1)\times\text{2x}$
$\Rightarrow5\text{x}^2+\text{5y}^2=\text{4x}^2+\text{y}^2+1-\text{4xy}+\text{2y}-\text{4y}$
$\Rightarrow5\text{x}^2+\text{5y}^2-\text{4x}^2-\text{y}^2-1+\text{4xy}-\text{2y}+\text{4y}=0$
$\Rightarrow\text{x}^2+\text{4y}^2+\text{4xy}+\text{4x}-\text{2y}-1=0$
This is the equetion of the required parabola.
View full question & answer→Question 35 Marks
Find the vertex, focus, axis, directrix and latus-rectum of the following parabolas:
$y^2 + 4x + 4y - 3 = 0$
AnswerThe given equation is
$\text{y}^2+\text{4x}+\text{4y}-3=0$
$\Rightarrow\text{y}^2+\text{4y}=-\text{4x}+3$
$\Rightarrow\text{y}^2+2\times\text{y}\times2+2^2=-\text{4x}+3+2^2$
$\Rightarrow\text{(y}+\text{2})^2=-\text{4x}+3+4$
$\Rightarrow\text{(y}+\text{2})^2=-\text{4x}+7$
$\Rightarrow\text{(y}+\text{2})^2=-4\Big(\text{x}-\frac{7}{4}\Big).....\text{(i)}$
Shifting the origin to the point $\Big(\frac{7}{4},-2\Big)$ without rotating the axes and denoting the new coordinates with respect to these axes by $X$ and $Y,$ we have
$\text{x}=\text{X}+\frac{7}{4},\ \text{y}=\text{Y}-2.....\text{(ii)}$
Using these relation (i), reduces to $\text{Y}^2=-\text{4X}.....\text{(iii)}$
This is of the form $\text{Y}^2=-4\text{X}$ on comparing, we get $\text{a} = 1$
Now,
Vertex: The coordinates of the vertex w.r.t new axes are $(\text{X}=0,\ \text{Y}=0)$
$\therefore\ \text{x}=0+\frac{7}{4},\ \text{y}=0-2$ [Using equation (ii)]
$\Rightarrow\text{x}=\frac{7}{4},\ \text{y}=-2$
$\therefore$ coordinates of the vertex w.r.t old axes are $\Big(\frac{7}{4},-2\Big).$
Focus: The coordinate of the focus w.r.t new axes are $(\text{X}=-1,\ \text{y}=0)$
$\therefore\text{ x}=-1+\frac{7}{4}$ and $\text{y}=0-2$ [Using equation (ii)]
$\Rightarrow\text{x}=\frac{3}{4},$ and $\text{y}=-2$
$\therefore$ coordinate of the focus w.r.t old axes are $\Big(\frac{3}{4},-2\Big).$
Axis: Equation of the axis of the parabola w.r.t new axes is
$\text{y}=0$
$\therefore\text{ y}=0-2$
$\Rightarrow\text{ y}=-2$
$\therefore$ equation of the w.r.t old axes is $\text{y}+2=0.$
View full question & answer→Question 45 Marks
Find the vertex, focus, axis, directrix and latus-rectum of the following parabolas:
$x^2 + y = 6x - 14.$
AnswerThe given equation is
$\text{x}^2 +\text{y} = \text{6x}-14$
$\Rightarrow\text{x}^2-\text{6x} =-\text{y} -14$
$\Rightarrow\text{x}^2-2\times\text{x}\times3+9 =-\text{y} -14+9$
$\Rightarrow\text{(x}-\text{3})^2 =-\text{y} -5$
$\Rightarrow\text{(x}-\text{3})^2 =-1\text{(y} +5).....\text{(i)}$
Shifting the origin to the point (3, -5) without rotating the axes and denoting the new coordinates w.r.t these axes by X and Y, we have,
$\text{x}=\text{X}+3,\ \text{y}=\text{Y}-5....\text{(ii)}$
Using these relation equation (i), redus to
$\text{X}^2=-\text{y}......\text{(iii)}$
This is of the form $\text{Y}=-\text{4aX},$ on comparing, we get
$\text{4a}=1$
$\Rightarrow\text{a}=\frac{1}{4}$
Now,
Vertex: The coordinates of the vertex w.r.t new axes are $(\text{X}=0,\ \text{Y}=0)$
$\therefore \text{ x}=0+3,\ \text{y}=0-5$
$\Rightarrow\text{x}=3,\ \text{y}=-5$
$\therefore$ coordinate of the vertex w.r.t new axes are $(3, -5).$
Focus: The coordinate of the focus w.r.t new axes are $\Big(\text{x}=0,\ \text{y}=\frac{-1}{4}\Big)$
$\therefore\text{x}=0+3,\ \text{y}=\frac{-1}{4}-5$
$\Rightarrow\text{x}=3,\ \text{y}=\frac{-21}{4}$
$\therefore$ Coordinates of the focus w.r.t old axes are $\Big(3,\frac{-21}{4}\Big)$
Axis: Equation of the axes of the parabola w.r.t new axes is
$\text{x}=0$
$\therefore\ \text{x}=0+3$
$\Rightarrow\text{x}=3$
$\therefore$ equation of axis w.r.t old axes is $\text{x}=3.$
Directrix: Equation of the directrix of the parabola w.r.t new axes is
$\text{y}=\frac{1}{4}$
$\therefore\ \text{y}=\frac{1}{4}-5$
$\Rightarrow\text{y}=\frac{-19}{4}$
$\Rightarrow\text{4y}+19=0$
$\therefore $ Equation of the directrix of the parabola w.r.t oid axes is $\text{4y}+19=0$
Latus-rectum: The length of the latus-rectum $= 4a$
$=4\times\frac{1}{4}$
$=1.$
View full question & answer→Question 55 Marks
If the points (O, 4) and(O, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.
AnswerThe vertex and focus of the parabola are A(0, 4) and F(0, 2) respectively.
$\text{AF} = 2$
As point A and F lie on y-axis, so y-axis is the axis of the parabola.
If the diretrix meets the axis of parabola at point Z, then $\text{AZ = AF} = 2.$
$\therefore\ \text{OZ = OF + FA + AZ}=2+2+2=6$
Let P(x, y) be any point in the plane of focus and diretrix, and MP be the perpendicular distance from P to the diretrix, then P lies on parabola iff
$\text{ FP = MP}$
$\Leftrightarrow\sqrt{\text{(x}-0)^2+\text{(y}-2)^2}=\frac{\big|\text{y}-6\big|}{1}$
$\Leftrightarrow\ \text{x}^2+\text{(y}-2)^2=\text{(y}-6)^2$
$\Leftrightarrow\ \text{x}^2+\text{y}^2-\text{4y}+4=\text{y}^2-\text{12y}+36$
$\Leftrightarrow\text{x}^2+\text{8y}=32$
$\text{x}^2+\text{8y}=32$ is the required equation of the parabola.
View full question & answer→Question 65 Marks
Find the vertex, focus, axis, directrix and latus-rectum of the following parabolas:
$y^2 - 4y + 4x = 0.$
AnswerThe given equation is
$\text{y}^2-\text{4y}+\text{4x}=0$
$\Rightarrow\text{y}^2-\text{4y}=-\text{4x}$
$\Rightarrow\text{y}^2-2\times\text{x}\times2+(2)^2=-\text{4x}+(2)^2$
$\Rightarrow\text{(y}-\text{2})^2=-\text{4x}+4$
$\Rightarrow\text{(y}-\text{2})^2=-\text{4(x}-1).....\text{(i)}$
Shifting the origin to the point (1, 2) without ratating the axes and denoting the new coordinates with respect to these axes by X and Y, we have
$\text{x}=\text{X}+1,\text{y}=\text{Y}+2.....\text{(ii)}$
Using these relations equation (i), reduces to
$\text{Y}^2=-\text{4X}....\text{(iii)}$
This is of the form $\text{Y}^2=-4\text{aX}.$
On comparing, we get, a = 1.
Now,
Vertex: The coordinates of the vertexw.r.t to new axes are $(\text{X}=0,\ \text{y=0}).$
$\therefore\text{ x}=0+1,\text{y}=0+2$ [using equation (ii)]
$\Rightarrow \text{x}=1,\text{y}=2$
$\therefore$ coordinates of the vertex w.r.t old axes are, (1, 2)
Focus: the coordinates of the focus with respect to new axes are $(\text{X}=0,\ \text{Y}=0).$
Putting $\text{X}=-1$ and $\text{Y}=0$ in equation (ii),we get
$\text{x}=-1+1,\ \text{y}=0+2$
$\Rightarrow\text{x}=0,\ \text{y}=2$
$\therefore$ coordinates of the focus w.r.t old axes are, (0, 2)
Axis: Equation of the axis of the parabola w.r.t new axes is $\text{Y}=0$
$\therefore \text{y}=0+2$ [Using equation (ii)]
$\Rightarrow\text{y}=2$
$\therefore$ equation of axis w.r.t old axes is $\text{y}=2$
Directrix: Equation of the directrix of the parabola w.r.t new axes is $\text{x}=1$
$\therefore\text{x}=1+1$ [Using equation (ii)]
$\Rightarrow\text{x}=2$
$\therefore$ equation of the directrix of the parabola w.r.t old axes is $\text{x}=2$
Latus-rectum: The length of the latus-rectum = 4a
$=4\times1$
$=4.$
View full question & answer→Question 75 Marks
The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100m long is supported by vertical wires attached to the cable, the longest wire being 30m and the shortest wire being 6m. Find the length of a supporting wire attached to the roadway 18m from the middle.
AnswerLet CAB be the bridge and LOX be the road way. Let A be the centre of the bridge. We find that the coordinates of A are (o, 6).
Clearly, the bridge is in the shope of a parabola having its vertex at A (D,6).
Let its equation be $\text{x}^2=\text{4a}\text{(y}-6)$
It posses through $\text{B}(50, 30).$ Therefore, $(50)^2=\text{4a}(30-6)$
$\Rightarrow\ 2500=\text{4a}\times24$
$\Rightarrow\frac{2500}{4\times24}=\text{a}$
$\Rightarrow\ \text{a}=\frac{625}{24}$
Putting the value of in (i), we get
$\text{x}^2=4\times\frac{625}{24}(\text{y}-6)$
$\Rightarrow\ \text{x}^2=\frac{625}{6}\text{(y}-6)$
Let l metres be the length of the vertical supporting cable 18 metres from the centre. Then, P(18, l) lies on (ii). Therefore
$(18)^2=\frac{625}{6}\text{(l}-6)$
$\Rightarrow\ 324\times6=625\text{(l}-6)$
$\Rightarrow\ \frac{1944}{625}=\text{(l}-6)$
$\Rightarrow\ \frac{1944}{625}+6=\text{l}$
$\Rightarrow\ \frac{1944+3750}{625}=\text{l}$
$\Rightarrow\ \text{l}=\frac{1944}{625}=9.11\text{m}\text{ (approx)}$
Hence, the required length of a supporting wire is 9.11m.
View full question & answer→Question 85 Marks
Find the equation of the parabola, if
The focus is at (-6, - 6) and the vertex is at (-2, 2).
AnswerGiven Focus (-6, -6)
Vertex (-2, 2)
Slope of lone connecting vertext and focuse is $\frac{2+6}{-2+6}=2$
Slope is the midpoint of focus and point on directrix which passes through axis
$-2=\frac{-6+\text{x}}{2};2=\frac{-6+\text{y}}{2}$
$(\text{x, y})=(2, 10)$
Equation of directrix is given by
$\text{y}-10=\frac{-1}{2}(\text{x}-2)$
$\text{2y}-20=-\text{x}=2$
$\text{x}+\text{2y}=22$
Equation of parabola is $\text{(x + 6})^2+(\text{y}+6)^2=\frac{\text{(x}+\text{2y}-22)^2}{5}$
$5\Big[\text{x}^2+\text{y}^2+36+36+\text{12x}+\text{12y}\Big]=\Big[\text{x}^2+\text{4y}^2+484+4\text{xy}-88\text{y}-\text{44x}\Big]$
$\text{4x}^2+\text{y}^2-124-\text{4xy}+\text{10x}+\text{148y}=0$
$\text{(2x}-\text{y})^2+4\text{(26x}+\text{37y}-31)=0.$
View full question & answer→Question 95 Marks
Find the equation of the parabola, if
The focus is at $(0, -3)$ and the vertex is at $(0, 0).$
AnswerIn a parabola, vertex is the mid-point of the focus and the point of the intersection of the axis and directrix. So, let $(x_1, y_1)$ be the co-ordinate of the point of intersection of the axis and directrix.
Then (0, 0) is the mid-point of the line segment joining $(0, 3)$ and $(x_1, y_1).$
$\therefore\frac{\text{x}_1+0}{2}=0$ and $\frac{\text{y}_1+0}{2}=0$
$\Rightarrow\text{x}_1=0$ and $\text{y}_1=3$
Thus, the directrix meets the axis at $(0, 3)$
$\therefore$ The equation of the directrix is $\text{y} = 3$
Clearly, the required parabola is not the form $\text{x}^2 = -\text{4ay},$ where $\text{a} = 3$
$\therefore$ equation of parabola is $\text{x}^2 = -4\times3\times\text{y}$
$\Rightarrow\text{x}^2=-\text{12y.}$
View full question & answer→Question 105 Marks
Find the coordinates of points on the parabola $y^2 = Bx$ whose focal distance is $4.$
AnswerWe have $\text{y}^2=\text{8x}$
$\Rightarrow\ \text{y}^2=\text{4(2)x}$
Comparing it with the genaral equation of parabola $\text{y}^2=\text{4ax},$ we will get $\text{a}=2$
Let the required point be $(x_1, y_1)$
Now, Focal distance $= 4$
$\Rightarrow\ \text{x}_1+\text{a}=4$
$\Rightarrow\ \text{x}_1+2=4$
$\Rightarrow\ \text{x}_1=4$
Now, the point will satisfy the equation of parabota
$\therefore(\text{y}_1)^2=8(2)=16$
$\Rightarrow\ \text{y}_1=\pm4$
Hence, the coordinate of the point are $(2, 4)$ and $(2, -4)$.
View full question & answer→Question 115 Marks
Find the equation of the parabola whose:
Focus is (1, 1) and the directrix is x + y + 1 = 0
AnswerLet P(X, Y) be any point on the parabola whose focus is S (1, 1) and the directrix x + y = 0. Draw PM perpendicular from P(X, Y) on the directrix x + y + 1 = 0.
Then by definition
$\text{SP}=\text{PM}$
$\Rightarrow\text{SP}^2=\text{PM}^2$
$\Rightarrow\text{(x}-1)^2+\text{(y}-1)^2=\Big(\frac{\text{x}+\text{y}+1}{\sqrt{1^2+1^2}}\Big)^2$
$\Rightarrow\text{x}^2+1-\text{2x}+\text{y}^2+1-\text{2y}=\Big(\frac{\text{x}+\text{y}+1}{\sqrt2}\Big)^2$
$\Rightarrow\text{x}^2+\text{y}^2-\text{2x}-\text{2y}+2=\frac{\text{(x}+\text{y}+1)^2}{2}$
$\Rightarrow\text{2(x}^2+\text{y}^2-\text{2x}-\text{2y}+2)=\text{x}^2+\text{y}^2+1+\text{2xy}+\text{2x}+\text{2y}$
$\Rightarrow\text{2x}^2+\text{2y}^2-\text{4x}-\text{4y}+4=\text{x}^2+\text{y}^2+1+\text{2xy}+\text{2x}+\text{2y}$
$\Rightarrow\text{2x}^2-\text{x}^2+\text{2y}^2-\text{y}^2-\text{2xy}-\text{4x}-\text{2x}-\text{4y}-\text{2y}+4-1=0$
$\Rightarrow\text{x}^2+\text{y}^2-\text{2xy}-\text{6x}-\text{6y}+3=0$
This is the equetion of the required parabola.
View full question & answer→Question 125 Marks
Find the area of the triartgle formed by the lines joining the vertex of the parabola $x^2 = 12y$ to the ends of its latus-rectum.
AnswerThe given equation of the parabola is $x^2 = 12y.$
On comparing the given equation with $x^2 = 4ay:$
$a = 3$
Required area $=\frac{1}{2}(\text{LL}'\times\text{OS}) =\frac{1}{2}\times12\times3=18 $ square units View full question & answer→Question 135 Marks
For the parabola $y^2 = 4px$ find the extremitiJs of a double ordinate of length 8p. Prove that the lines from the vertex to its extremities are at right angles.
AnswerLet $PQ$ be the double ordinate of length 8p of the parabola $\text{y}^2==\text{4px}.$
Then, $PR = QR = 4p.$
Let $\text{AR} = \text{x}_1.$ Then the coordinate of P and Q are $(\text{x}_1,\ 4\text{p})$ and $(\text{x}_1,\ -4\text{p})$ respectively.
Since P line on $\text{y}^2=\text{4px}$
$\therefore\ \text{(4p)}^2=\text{4px}_1$
$\Rightarrow\ \text{x}_1=4\text{p}$
So, coordinates of p and Q are $\text{(4p,}\text{ 4p})$ and $\text{(4p,}\ -\text{4p})$ respectively.
$\therefore$ The extremities of a double ordinate are $\text{(4p,}\text{ 4p})$ and $\text{(4p,}\ -\text{4p}).$
Also, the coordinates of the vertex $A$ are $(o, o).$
$\therefore$ m_1 = slope of AP
$=\frac{4\text{p}-0}{4\text{p}-0}$
$=1$
and, $m_2 =$ slope of $\text{AQ}=\frac{-\text{4p}-0}{\text{4p}-0}$
$=-1$
Clear $\text{y, m}_1\text{m}_2=-1.$
Hence, $\text{AP}\bot\text{AQ}$
$\therefore$ The lines from the vertex to its extremities are at right angles.
View full question & answer→Question 145 Marks
Find the equation of the parabola whose:
Focus is (3, 0) and the directrix is 3 x + 4y = 1
AnswerLet P(X, Y) be any point on the parabola whose focus is S(3, 0) and the directrix 3x + 4y = 1. Draw PM perpendicular from P(X, Y) on the diretrix 3x + 4y = 1.
Then by definition
$\text{SP = PM}$
$\Rightarrow\text{SP}^2=\text{PM}^2$
$\Rightarrow\text{(x}-3)^2+\text{(y}-0)^2=\Big(\frac{\text{3x+4y}-1}{\sqrt{(3)^2+(4)^2}}\Big)^2$
$\Rightarrow\text{x}^2+9-6\text{x}+\text{y}^2=\Big(\frac{\text{3x+4y}-1}{\sqrt{9+16}}\Big)^2$
$\Rightarrow\text{x}^2+9-\text{6x}+\text{y}^2=\frac{\text{(3x+4y}-1)^2}{\big(\sqrt{25}\big)^2}$
$\Rightarrow\text{x}^2+9+\text{6x}+\text{y}^2=\frac{\text{(3x}+\text{4y}-1)^2}{25}$
$\Rightarrow25(\text{x}^2-\text{6x}+\text{y}^2+9)=(\text{3x}+\text{4y}-1)^2$
$\Rightarrow25\text{x}^2-\text{150x}+\text{25y}^2+225=\text{(3x)}^2+\text{(4y)}^2+(-1)^2\\+2\times\text{3x}\times\text{4y}+2\times\text{4y}\times(-1)+2\times(-1)\times\text{3x}$
$\Rightarrow\text{25x}^2-\text{150x}+\text{25y}^2+225=\text{9x}^2+\text{16y}^2+1+\text{24xy}-\text{8y}-\text{6x}$
$\Rightarrow\text{25x}^2-\text{9x}^2+\text{25y}^2-\text{16y}^2-\text{150x}+\text{6x}+\text{8y}-\text{24xy}+225=0$
$\Rightarrow\text{16x}^2+\text{9x}^2-\text{144x}+\text{8y}-\text{24xy}+224=0$
$\Rightarrow\text{16x}^2+\text{9x}^2-\text{24xy}-\text{144x}+\text{8y}+224=0$
This is the questions of the required parabola.
View full question & answer→Question 155 Marks
Find the equations of the lines joining the vertex of the parabola $y^2 = 6x$ to the point on it which have abscissa 24.
AnswerLet $A$ and $B$ be points on the parabola $y^2 = 6x$ and $OA, OB$ be the lines joining the vertex $O$ to the points $A$ and $B$ whose abscissa are $24.$
Now, $y^2 = 6 \times 24 = 144 y^2 = \pm 12$
Therefore the coordinates of the points $A$ and $B$ are $(24, 12)$ and $(24, -12)$ respectively. Hence the lines are given by $\text{y}-0=\pm\ \frac{12-0}{24-0}(\text{x}-0)$ $\Rightarrow\pm\ 2\text{y}=\text{x}$ View full question & answer→Question 165 Marks
If the line $y = mx + 1$ is tangent to the parabola $y^2 = 4x,$ then find the value of $m.$
AnswerThe line $y = mx + 1$ is tangent to the parabola $y^2 = 4x.$
$\therefore\text{(mx}=1)^2=4\text{x}$
$\text{m}^2\text{x}^2+\text{(4mx}+1)=\text{4x}$
$\text{m}^2\text{x}^2+\text{(2m}-1)\text{x}+1=0$
As we know tangent touches the parabola, so the roots of the above quadratic wi II be equal.
$\Rightarrow\text{D}=\text{b}^2-\text{4ac}=0$
$\Rightarrow\ (2\text{m}-4)^2-4\text{(m}^2)(1)=0$
$\Rightarrow\text{4m}^2-16+\text{16m}-\text{4m}^2=0$
$\Rightarrow\text{m}=1.$
View full question & answer→Question 175 Marks
Find the equation of the parabola whose:
Focus is (2, 3) and the directrix x - 4 y + 3 = 0
AnswerLet P(X, Y) be any point on the parabola whose focus is S (2, 3) and the directrix x - 4y + 3 = 0. Draw PM perpendicular from P(X, Y) on the directrix x - 4y + 3 = 0.
Then by definition
$\text{SP}=\text{PM}$
$\Rightarrow\text{SP}^2=\text{PM}^2$
$\Rightarrow\text{(x}-2)^2+\text{(y}-3)^2=\Bigg(\frac{\text{x}-\text{4y}+3}{\sqrt{1^2+(-4)^2}}\Bigg)^2$
$\Rightarrow\text{x}^2+4-\text{4x}+\text{y}^2+9-\text{6y}=\frac{\text{(x}-\text{4y}+3)^2}{(\sqrt{17})^2}$
$\Rightarrow\text{x}^2+\text{y}^2-\text{4x}-\text{6y}+4+9=\frac{(\text{x}-\text{4y}+3)^2}{17}$
$\Rightarrow\text{17(x}^2+\text{y}^2-\text{4x}-\text{6y}+13)=\text{(x}-\text{4y}+3)^2$
$\Rightarrow\text{17x}^2+\text{17y}^2-\text{68x}-\text{102y}+122=\text{x}^2+(-\text{4y})^2\\+3^2+2\times\text{x}\times(-\text{4y)}+2\times(-\text{4y)}\times3+2\times3\times\text{x}$
$\Rightarrow\text{17x}^2+\text{17y}^2-\text{68x}-\text{102y}+221=\text{x}^2+\text{16y}^2+9-\text{8xy}-\text{24y}+\text{6x}$
$\Rightarrow\text{17x}^2-\text{x}^2+\text{17y}^2-\text{16}^2+\text{8xy}-\text{68x}-\text{6x}-\text{102y}+\text{24y}+221-9=0$
$\Rightarrow\text{16x}^2+\text{y}^2+\text{8xy}-\text{74x}-\text{78y}+212=0$
This is the questions of the required parabola.
View full question & answer→Question 185 Marks
Find the equation of the parabola, if
The focus is at (0, 0) and vertex is at the intersection of the lines x + y = 1 and x - y = 3.
Answer$\text{x}+\text{y}=1$ and $\text{x}-\text{y}=3$
Intersecting point of above lines is
$\text{(x, y)}=(2,1)......\text{vertex}$
Focus (0, 0)
Vertex is the mid-piont of focus and point on directrix which passes through
$2=\frac{0+\text{x}}{2};-1=\frac{0+\text{y}}{2}$
$\text{(x, y)}=(2, 4)$
Slope of line passing through focus and vertex is $\frac{-1}{2}$
Slope of directrix is 2, as both are perpendicular lines
$\text{y}+2=2(\text{x}-4)$
$\text{2x}-\text{y}=10.....\text{directrix}$
$\text{SP}^2=\text{PM}^2$
$5(\text{x}^2+\text{y}^2)=(\text{2x}-\text{y}-10)^2$
$\text{x}^2+\text{4y}^2-100+\text{4xy}-\text{20y}+\text{40x}=0$
$(\text{x}+\text{2y})^2+20(\text{2x}-\text{y}-5)=0.$
View full question & answer→Question 195 Marks
Find the vertex, focus, axis, di$rectrix and latus-rectum of the following parabolas:$
$y^2 = 8x + 8y.$
AnswerThe given equation is
$\text{y}^2=\text{8x}+\text{8y}$
$\Rightarrow\text{y}^2-\text{8y}=\text{8x}$
$\Rightarrow\text{y}^2+2\times\text{y}\times4+16=\text{4x}+16$
$\Rightarrow\text{(y}-\text{4})^2=8(\text{x}+2).....\text{(i)}$
Shifting the origin to the point $(-2, 4)$ without rotating the axes and denoting the new coordinates w.r.t these axes by $X$ and $Y,$ we have
$\text{x}=\text{X}-2,\ \text{y}=\text{Y}+4.....\text{(ii)}$
Using these relation equation (i), reduces to $\text{Y}^2=\text{8X}.....\text{(iii)}$
This is of the form $\text{Y}^2=4\text{aX},$ on comparing, we get
$\text{4a} = 8$
$\Rightarrow\text{a}=2$
Now,
Vertex: The coordinates of the vertex w.r.t new axes are $(\text{X}=0,\ \text{Y}=0)$
$\therefore\ \text{x}=0-2,\ \text{y}=0+4$ [Using equation (ii)]
$\Rightarrow\text{x}=0-2,\ \text{y}=4$
$\therefore$ coordinates of the vertex w.r.t old axes are $\Big(-2,4\Big).$
Focus: The coordinate of the focus w.r.t new axes are $(\text{X}=2,\ \text{y}=0)$
$\therefore\text{ x}=-2-2$ and $\text{y}=0-4$ [Using equation (ii)]
$\Rightarrow\text{x}=0,$ and $\text{y}=4$
$\therefore$ coordinate of the focus w.r.t old axes are $(0,4).$
Axis: Equation of the axis of the parabola w.r.t new axes is $\text{y}=0$
$\therefore\text{ y}=0+4$ [Using equation (ii)]
$\Rightarrow\text{ y}=4$
$\therefore$ equation of the w.r.t old axes is $\text{y}-4.$
Directrix; Equation of the parabola w.r.t new axes is
$\text{X}=-2$
$\therefore\ \text{x}=-2-2$
$\Rightarrow\text{x}=-4$
$\Rightarrow\text{x}+4=0$
$\therefore$ Equetion of the directrix of the parabola w.r.t old axes is $\text{x}+4=0$
Latus-rectum: The length of the latus-rectum $= 4a$
$=4\times2$
$=8.$
View full question & answer→Question 205 Marks
Find the equation of the parabola, if
The focus is at $(a, 0)$ and the vertex is at $(a', 0).$
AnswerIn a parabola, vertex is the mid-point of the focus and the point of the intersection of the axis and directrix. So, let $(x_1, y_1)$ be the co-ordinate of the point of intersection of the axis and directrix.
Then $(a', 0)$ is the mid-point of the line segment joining $(a, 0)$ and $(x_1, y_1).$
$\therefore\frac{\text{x}_1+\text{a}}{2}=\text{a}'$ and $\frac{\text{y}_1-0}{2}=0$
$\Rightarrow\text{x}_1=2\text{a}'-\text{a}$ and $\text{y}_1=0$
Thus, the directrix meets the axis at $(2a'-a, 0)$
So the equation of directrix is $x = 2a' - a$
Let P(x, y) be any point on the parabola.
Then,
$\text{SP}=\text{PM}$
$\Rightarrow\text{SP}^2=\text{PM}^2$
$\Rightarrow\text{(x}-\text{a)}^2+\text{(y}-0)^2=\Big[\frac{\text{x}-\text{2a}'+\text{a}}{\sqrt{1^2}}\Big]^2$
$\Rightarrow\text{x}^2+\text{a}^2-\text{2ax}+\text{y}^2=\text{(x}-\text{2a}'+\text{a})^2$
$\Rightarrow\text{x}^2+\text{a}^2-\text{2ax}+\text{y}^2=\text{x}^2+(-\text{2a}')^2+\text{a}^2+\\2\times\text{x}\times(-\text{2a}')+2\times(-\text{2a}')\times\text{a}+2\times\text{(a)}\times\text{(x)}$
$\Rightarrow\text{x}^2+\text{a}^2-\text{2ax}+\text{y}^2=\text{x}^2+4\text{(a}')^2+\text{a}^2-\text{4xa}'-\text{a}'\text{a}+\text{2ax}$
$\Rightarrow\text{y}^2=\text{x}^2-\text{x}^2+\text{a}^2-\text{a}^2+\text{2ax}+4\text{(a}')^2-\text{4xa}'-4\text{a}'\text{a}+\text{2ax}$
$\Rightarrow\text{y}^2=\text{4ax}-\text{4xa}'+\text{4(a}')\text{a}$
$\Rightarrow\text{y}^2=\text{4ax}-\text{4a}'\text{a}-\text{4xa}'+4\text{(a}')^2$
$=\text{4a(x}-\text{a}')-\text{4a}'\text{(x}-\text{a}')$
$=\text{(4a}-\text{4a}')\text{(x}-\text{a}')$
$=4\text{(a}-\text{a}')\text{(x}-\text{a}')$
$\therefore\text{y}^2=4\text{(a}-\text{a}')\text{(x}-\text{a}')$
$\Rightarrow\text{y}^2=-4\text{(a}'-\text{a})\text{(x}-\text{a}')$
Hence, required equation of parabola is $\text{y}^2=-4\text{(a}'-\text{a})\text{(x}-\text{a}').$
View full question & answer→Question 215 Marks
Find the coordinates of the point of intersection of the axis and the directrix of the parabola whose focus is (3, 3) and directrix is 3x - 4y = 2. Find also the length of the latus-rectum.
AnswerThe axis of the parabola Is a line $\bot$ to the directrix and passing through focus. The equation of a line $\bot$ to $\text{3x}-\text{4y}-2=0$ is
$\text{y}=\frac{-4}{3}+\lambda$ $\begin{bmatrix}\therefore\text{ m}_1\text{m}_2=-1\\\Rightarrow\text{m}_2\frac{-1}{\text{m}_1} \text{ and y =}\text{m}_2\text{x}+\lambda\end{bmatrix}$
$\Rightarrow\ \text{3y}+\text{4x}=3\lambda$
This will pass through focus (3,3) if,
$3\times3+4\times3=3\lambda$
$\Rightarrow\ 9+12=3\lambda$
$\Rightarrow\ 21=3\lambda$
$\Rightarrow\ \lambda=\frac{21}{3}=7$
So, the equation of axis is $\text{3y}+\text{4x}=3\times7=21$
$\Rightarrow\ \text{3y}+\text{4x}=21...\text{(i)}$
And the equation of doreanx is
$\text{3x}-\text{4y}=2....\text{(ii)}$
Mutiplying equation (i) by 4 and equation (ii) by 3, we get
$\text{16x}+\text{12y}=84.....\text{(iii)}$
$\text{9x}-\text{12y}=6....\text{(iv)}$
Adding equation (iii) and (iv), we get
$\text{16x}+\text{9x}=84+6$
$\Rightarrow\ 25\text{x}=90$
$\Rightarrow\ \text{x}=\frac{90}{25}=\frac{18}{5}$
Putting $\text{x}=\frac{18}{5}$ in equation (i), we get
$\text{3y}+4\times\frac{18}{5}=21$
$\Rightarrow\ \text{3y}+\frac{72}{5}=21$
$\Rightarrow\ \text{3y}=21-\frac{72}{5}$
$\Rightarrow\ \text{3y}=\frac{105-72}{5}$
$\Rightarrow\ \text{3y}=\frac{33}{5}$
$\Rightarrow\ \text{y}=\frac{11}{5}$
Hence, the required point of intersection is $\Big(\frac{18}{5},\frac{11}{5}\Big).$
View full question & answer→Question 225 Marks
Find the vertex, focus, axis, directrix and latus-rectum of the following parabolas:
$4(y - 1)^2 = -7(x - 3).$
AnswerThe given equation is
$4(\text{y}-1)^2=-7(\text{x}-3)$
$\Rightarrow(\text{y}-1)^2=\frac{-7}{4}(\text{x}-3)...\text{(i)}$
Shifting the origin to the point $(3, 1)$ without rotating the axes and denoting the new coordinates w.r.t these axes by $X$ and $Y,$ we have,
$\text{x}=\text{X}+3,\ \text{y}=\text{Y}+1....\text{(ii)}$
Using these relation (i), redus to
$\text{Y}^2=\frac{-7}{4}\text{X}......\text{(iii)}$
This is of the form $\text{Y}=-\text{4aX},$ on comparing, we get
$\text{4a}=\frac{7}{4}$
$\Rightarrow\text{a}=\frac{7}{16}$
Now,
Vertex: The coordinates of the vertex w.r.t new axes are $(\text{X}=0,\ \text{Y}=0)$
$\therefore \text{ x}=0+3,\ \text{y}=0+1$ [Using equation (iii)]
$\Rightarrow\text{x}=3,\ \text{y}=1$
$\therefore$ coordinate of the vertex w.r.t new axes are $(3, 1).$
Focus: The coordinate of the focus w.r.t new axes are $\Big(\text{x}=-\frac{7}{16},\text{y}=0\Big)$
$\therefore\text{x}-\frac{-7}{16}+3,\ \text{y}=0+1$
$\Rightarrow\text{x}=\frac{41}{16},\ \text{y}=1$
$\therefore$ coordinates of the focus w.r.t old axes are $\Big(\frac{41}{16},1\Big)$
Axis: Equation of the axes of the parabola w.r.t new axes is
$\text{y}=0$
$\Rightarrow\text{y}=0+1$
$\Rightarrow\text{y}=1$
$\therefore$ equation of axis w.r.t old axes is $\text{y}=1$
Directrix: Equation of the directrix of the parabola w.r.t new axes is
$\text{Y}=\frac{7}{16}$
$\therefore\text{x}=\frac{7}{16}+3$
$\Rightarrow\text{x}=\frac{55}{16}$
$\therefore $ Equation of the directrix of the parabola w.r.t oid axes is $\text{x}=\frac{55}{16}.$
Latus-rectum: The length of the latus-rectum $= 4a$
$=4\times\frac{7}{16}$
$=\frac{7}{4}.$
View full question & answer→Question 235 Marks
Find the vertex, focus, axis, directrix and latus-rectum of the following parabolas:
$4x^2 + y = 0.$
AnswerIn the given parabola, $\text{a}=\frac{1}{16}$
Focus $(0,\frac{1}{16})$
vertex $(0, 0)$
Directrix $\text{x,y}=\frac{1}{16}$
Axis, $\text{x} = 0$
$\text{LR}=\frac{1}{4}$
The given equation is
$\text{y}^2-\text{4y}-\text{3x}+1=0$
$\Rightarrow\text{y}^2-\text{4y}=\text{3x}-1$
$\Rightarrow\text{y}^2-\text{4y}+4=\text{3x}-1+4$
$\Rightarrow\text{y}^2-\text{4y}+(2)^2=\text{3x}+3$
$\Rightarrow\text{(y}-\text{2})^2=3\text{(x}+1)....\text{(i)}$
Shifting the origin to the point $(-1, 2)$ without rotating the axis and denote the new coordinates with respect to these axis by $X$ and $Y,$ we have
$\text{x}=\text{X}-1,\text{y}=\text{Y}+2....\text{(ii)}$$$
Using these relations equation (i), reduces to
$\text{y}^2=\text{3X}...\text{(iii)}$
This is of the form $\text{y}^2=\text{4aX}.$
On comparing we get,
$\text{4a}=3$
$\Rightarrow\text{a}=\frac{3}{4}.$
Now,
Vertex: The coordinate of the vertex with respect to new axes are $\text{X}=0,\ \text{Y}=0$
So coordinates of the vertex with respect to old axes are $(-1, 2)$
Focus: The coordinates of the focus with respect to new axes are $\Big(\text{X}=\frac{3}{4},\ \text{Y}=0\Big).$
Putting $\text{X}=\frac{3}{4}$ and $\text{Y}=0$ in equation (ii), we get
$\text{X}=\frac{3}{4}-1$ and $\text{y}=0+ 2$
$\Rightarrow\text{x}=\frac{-1}{4}$ and $\text{y}=2$
$\therefore$ coordinates of the focus of the with respects to old axe are $\Big(\frac{-1}{4},2\Big).$
View full question & answer→Question 245 Marks
Find the equation of the parabola, if
The focus is at $(0, -3)$ and the vertex is at $(-1, -3).$
AnswerIn a parabola, vertex is the mid-point of the focus and the point of the intersection of the axis and directrix. So, let $(x_1, y_1)$ be the co-ordinate of the point of intersection of the axis and directrix.
Then $(-1, -3)$ is the mid-point of the line segment joining $(0, -3)$ and $(x_1, y_1).$
$\therefore\frac{\text{x}_1+0}{2}=-1$ and $\frac{\text{y}_1-3}{2}=-3$
$\Rightarrow\text{x}_1=-2$ and $\text{y}_1=-3$
Thus, the directrix meets the axis at $(-2, -3)$
Let A be the vertex and S be the focus of the required parabola.
Then,
$\text{m}_1=\text{slope of AS}=\frac{-3-(-3)}{0-(-1)}=0$
$\therefore$ slope of the directrix $=\frac{-1}{0}=\infty$
Thus, the directrix passes through (-2, -3) and has slope $\infty,$ so its equation is
$\text{y}-(-3)=\infty(x-(-2))$
$\frac{\text{y}+3}{\infty}=\text{x}+2$
$\Rightarrow\text{x}+2$
Let $P(x, y)$ be a point on parabola.
Then, $PS =$ Distance of P from the directrix.
$\sqrt{\text{(x}-2)^2+\text{(y}+3)^2}=\Big|\frac{\text{x}+2}{\sqrt{1^2}}\Big|$
$\Rightarrow\text{x}^2+\text{(y}+3)^2=\text{(x}+2)^2$
$\Rightarrow\text{x}^2+\text{y}^2+9+\text{6y}=\text{x}^2+4+\text{4x}$
$\Rightarrow\text{y}^2-\text{4x}+\text{6y}+9-4=0$
$\Rightarrow\text{y}^2-\text{4x}+\text{6y}+5=0.$
View full question & answer→Question 255 Marks
Find the equation of the parabola whose focus is the point (2, 3) and directrix is the line x - 4y + 3 = 0. Also, find the length of its latus-rectum.
AnswerLet P(X, Y) be any point on the parabola whose focus is S (2, 3) and the directrix x - 4y + 3 = 0. Draw PM perpendicular from P(X, Y) on the directrix x - 4y + 3 = 0.
Then by definition
$\text{SP}=\text{PM}$
$\Rightarrow\text{SP}^2=\text{PM}^2$
$\Rightarrow\text{(x}-2)^2+\text{(y}-3)^2=\Bigg(\frac{\text{x}-\text{4y}+3}{\sqrt{1^2+(-4)^2}}\Bigg)^2$
$\Rightarrow\text{x}^2+4-\text{4x}+\text{y}^2+9-\text{6y}=\frac{\text{(x}-\text{4y}+3)^2}{(\sqrt{17})^2}$
$\Rightarrow\text{x}^2+\text{y}^2-\text{4x}-\text{6y}+4+9=\frac{(\text{x}-\text{4y}+3)^2}{17}$
$\Rightarrow\text{17(x}^2+\text{y}^2-\text{4x}-\text{6y}+13)=\text{(x}-\text{4y}+3)^2$
$\Rightarrow\text{17x}^2+\text{17y}^2-\text{68x}-\text{102y}+122=\text{x}^2+(-\text{4y})^2\\+3^2+2\times\text{x}\times(-\text{4y)}+2\times(-\text{4y)}\times3+2\times3\times\text{x}$
$\Rightarrow\text{17x}^2+\text{17y}^2-\text{68x}-\text{102y}+221=\text{x}^2+\text{16y}^2+9-\text{8xy}-\text{24y}+\text{6x}$
$\Rightarrow\text{17x}^2-\text{x}^2+\text{17y}^2-\text{16}^2+\text{8xy}-\text{68x}-\text{6x}-\text{102y}+\text{24y}+221-9=0$
$\Rightarrow\text{16x}^2+\text{y}^2+\text{8xy}-\text{74x}-\text{78y}+212=0$
This is the questions of the required parabola.
Latus Rectum = Legnth of perprndicular from focus (2, 3) on directrix x - 4y + 3 = 0
$=2\Big|\frac{2-12+3}{\sqrt{1+16}}\Big|$
$=2\Big|\frac{-7}{\sqrt{17}}\Big|$
$=\Big|\frac{14}{\sqrt{17}}\Big|.$
View full question & answer→