Sample QuestionsConic Sections questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If the equation $\frac{\lambda(\text{x}+1)^2}{3}+\frac{(\text{y}+2)^2}{4}=1$ represents a circle then $\lambda$:
- A
$1$
- ✓
$\frac{3}{4}$
- C
$0$
- D
$-\frac{3}{4}$
Answer: B.
View full solution →The eccentricity of the ellipse, if the distance between the foci is equal to the length of the latus$-$rectum, is:
- ✓
$\frac{\sqrt{5}-1}{2}$
- B
$\frac{\sqrt{5}+1}{2}$
- C
$\frac{\sqrt{5}-1}{4}$
- D
$\text{none of these}$
Answer: A.
View full solution →Find the equation of the circle. Centered at $(3, -2)$ with radius $4:$
- A
$ x^2+y^2+6 x-4 y=3 $
- ✓
$ x^2+y^2-6 x+4 y=3 $
- C
$ x^2+y^2-3 x+2 y=-3 $
- D
Answer: B.
View full solution →The equation of ellipse whose one focus is at $(4, 0)$ and whose eccentricity is $\frac{4}{5}$ is:
- A
$\frac{\text{x}^2}{5}+\frac{\text{y}^2}{9}=1$
- ✓
$\frac{\text{x}^2}{25}+\frac{\text{y}^2}{9}=1$
- C
$\frac{\text{x}^2}{9}+\frac{\text{y}^2}{5}=1$
- D
$\frac{\text{x}^2}{9}+\frac{\text{y}^2}{25}=1$
Answer: B.
View full solution →The equation of the incircle formed by the coordinate axes and the line $4x + 3y = 6$ is:
Answer: B.
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
If the distances of foci and vertex of hyperbola from the centre are $c$ and $a$ respectively, then
Assertion: A line through the focus and perpendicular to the directrix is called the $x -$ axis of the parabola.
Reason: The point of intersection of parabola with the axis is called the vertex of the parabola.
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
Answer: D.
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The sum of focal distances of a point on the ellipse $9x^2 + 4y^2 - 18x - 24y + 9 = 0$ is $4.$
Reason: The equation $9x^2+ 4y^2 - 18x - 24y + 9 = 0$ can be expressed as $9(x -1)^2 + 4(y - 3)^2 = 36.$
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
Answer: D.
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The length of major and minor axes of the ellipse $5x^2 + 9y^2 - 54y + 36 = 0$ are $6$ and $10,$ respectively.
Reason: The equation $5x^2 + 9y^2 - 54y + 36 = 0$ can be expressed as $5x^2 + 9(y - 3)^2 = 45.$
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
Answer: D.
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
If the distances of foci and vertex of hyperbola from the centre are c and a respectively, then
Assertion: Eccentricity is always less than $1.$
Reason: Foci are at a distance of ae from the centre.
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
Answer: D.
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Parabola is symmetric with respect to the axis of the parabola.
Assertion: If the equation of standard parabola has a term $y^2,$ then the axis of symmetry is along the $x - $ axis.
Reason: If the equation of standard parabola has a term $x^2,$ then the axis of symmetry is along the $x -$ axis.
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- ✓
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
Answer: C.
View full solution →State Whether the statements are True or False. Justify.
The line $x + 3y = 0$ is a diameter of the circle $x^2 + y^2 + 6x + 2y = 0.$
View full solution →State Whether the statements are True or False. Justify.
The shortest distance from the point $(2, -7)$ to the circle $x^2 + y^2 - 14x - 10y - 151 = 0$ is equal to $5.$
[Hint: The shortest distance is equal to the difference of the radius and the distance between the centre and the given point]
View full solution →State Whether the statements are True or False. Justify.
The locus of the point of intersection of lines $\sqrt{3}\text{x}-\text{y}-4\sqrt{3}\text{k}=0$ and $\sqrt{3}\text{kx}+\text{ky}-4\sqrt{3}=0$ for different value of k is a hyperbola whose eccentricity is 2.
[Hint: Eliminate k between the given equations]
View full solution →State Whether the statements are True or False. Justify.
The line $lx + my + n = 0$ will touch the parabola $y^2 = 4ax$ if $ln = am^2$
View full solution →State Whether the statements are True or False. Justify.
If the line $lx + my = 1$ is a tangent to the circle $x^2 + y^2 = a^2,$ then the point $(l, m)$ lies on a circle.
[Hint: Use that distance from the centre of the circle to the given line is equal to radius of the circle]
View full solution →Find the equation of the circle with centre $(-3, 2)$ and radius $4$
View full solution →Find an equation of the circle with centre at $(0,0)$ and radius $r.$
View full solution →An equilateral triangle is inscribed in the parabola $y^2 = 4ax$ where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
View full solution →Find the area of the triangle formed by the lines joining the vertex of the parabola $x^2 = 12y$ to the ends of its latus rectum.
View full solution →If a parabolic reflector is $20\ cm$ in diameter and $5\ cm$ deep, find the focus.
View full solution →Find the equation of the hyperbola, whose vertices $(0, \pm 3)$ and foci $(0, \pm 5).$
View full solution →Find the equation of hyperbola having Vertices (0, $\pm$5), foci (0, $\pm$8)
View full solution →A man running a racecourse notes that the sum of the distances from the two flag posts from him is always $10$ m and the distance between the flag posts is $8$ m. Find the equation of the path traced by the man.
View full solution →A rod of length $12$ m moves with its ends always touching the coordinates axes. Determine the equation of the locus of a point P on the rod, which is $3$ cm from the end in contact with the $X$-axis.
View full solution →An arc is in the form of a semi-ellipse. It is $8$ m wide and $2$ m high at the centre. Find the height of the arch at a point $1.5$ m from one end.
View full solution →The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and $100$ m long is supported by vertical wires attached to the cable, the longest wire being $30$ m and the shortest being $6$ m. Find the length of a supporting wire attached to the roadway $18$ m from the middle.
View full solution →An arc is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it $2$ m from the vertex of the parabola?
View full solution →Fill in the blank.
The equation of the circle circumscribing the triangle whose sides are the lines $y = x + 2, 3y = 4x, 2y = 3x$ is ___________.
View full solution →Fill in the blank.
The equation of the ellipse having foci (0, 1), (0, -1) and minor axis of length 1 is _____________.
Fill in the blank.
The equation of the parabola having focus at $(-1, -2)$ and the directrix $x - 2y + 3 = 0$ is _____________.
View full solution →Fill in the blank.
The equation of the hyperbola with vertices at $(0,\pm6)$ and eccentricity $\frac{5}{3}$ is ___________ and its foci are ___________.
View full solution →Fill in the blank.
An ellipse is described by using an endless string which is passed over two pins. If the axes are 6cm and 4cm, the length of the string and distance between the pins are ____________.