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.
The eccentricity of the hyperbola whose latus rectum is $8$ and conjugate axis is equal to half of the distance between the foci is:
- A
$\frac{4}{3}$
- B
$\frac{4}{\sqrt{3}}$
- ✓
$\frac{2}{\sqrt{3}}$
- D
Answer: C.
View full solution →The equation of the ellipse whose focus is $(1, -1),$ the directrix the line $x - y - 3 = 0$ and eccentricity $\frac{1}{2}$ is:
- ✓
$7x^2 + 2xy + 7y^2 - 10x + 10y + 7 = 0$
- B
$7x^2 + 2xy + 7y^2 + 7 = 0$
- C
$7x^2 + 2xy + 7y^2 + 10x - 10y - 7 = 0$
- D
Answer: A.
View full solution →The area of the circle centred at $(1, 2)$ and passing through $(4, 6)$ is:
Answer: C.
View full solution →The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length $3a$ is:
$[$Hint: Centroid of the triangle coincides with the centre of the circle and the radius of the circle is $\frac{2}{3}$ of the length of the mediam$]$
- A
$x^2 + y^2 = 9a^2$
- B
$x^2 + y^2 = 16a^2$
- ✓
$x^2 + y^2 = 4a^2$
- D
$x^2 + y^2= a^2$
Answer: C.
View full solution →Equation of a circle which passes through $(3, 6)$ and touches the axes is:
- A
$x^2 + y^2 + 6x + 6y + 3 = 0$
- B
$x^2 + y^2 - 6x - 6y - 9 = 0$
- ✓
$x^2 + y^2 - 6x - 6y + 9 = 0$
- D
Answer: C.
View full solution →The line $x + 3y = 0$ is a diameter of the circle $x^2 + y^2 + 6x + 2y = 0.$
View full solution →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 →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 →The line $lx + my + n = 0$ will touch the parabola $y^2 = 4ax$ if ln $= am^2$
View full solution →If the line $lx + my = 1$ is a tangent to the circle $x^2 + y^2 = a^2$, then the point $\text{(l, m)}$ 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 eccentricity of the hyperbola $9y^2 - 4x^2 = 36.$
View full solution →Find the equation of the following parabolas:
Directrix $x = 0,$ focus at $(6, 0)$
View full solution →Find the equation of the circle which touches the both axes in first quadrant and whose radius is a.
Find the equation of the hyperbola with:
$\text{Vertices }(\pm5,0),\text{ foci }(\pm7,0)$
View full solution →If the latus rectum of an ellipse is equal to half of minor axis, then find its eccentricity.
Given the ellipse with equation $9x^2 + 25y^2 = 225$, find the eccentricity and foci.
View full solution →Find the equation of a circle passing through the point $(7, 3)$ having radius $3$ units and whose centre lies on the line $y = x - 1.$
View full solution →Find the equation of the circle which touches $x-$axis and whose centre is $(1, 2).$
View full solution →Find the equation of the circle which passes through the points $(2, 3)$ and $(4, 5)$ and the centre lies on the straight line $y - 4x + 3 = 0.$
View full solution →Show that the point (x, y) given by $\text{x}=\frac{2\text{at}}{1+\text{t}^2}$ and $\text{y}=\frac{\text{a}(1-\text{t}^2)}{1+\text{t}}$ lies on a circle for all real values of t such that $-1\leq\text{y}\leq1$ where a is any given real numbers.
View full solution →If the line $y = mx + 1$ is tangent to the parabola $y^2 = 4x$ then find the value of m.
$[$Hint: Solving the equation of line and parabola, we obtain a quadratic equation and then apply the tangency condition giving the value of m$]$
View full solution →The equation of the circle circumscribing the triangle whose sides are the lines $y = x + 2, 3y = 4x, 2y = 3x$ is ___________.
View full solution →The equation of the ellipse having foci (0, 1), (0, -1) and minor axis of length 1 is _____________.
The equation of the parabola having focus at $(-1, -2)$ and the directrix $x - 2y + 3 = 0$ is _____________.
View full solution →The equation of the hyperbola with vertices at $(0,\pm6)$ and eccentricity $\frac{5}{3}$ is ___________ and its foci are ___________.
View full solution →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 ____________.
Find the equation of a circle concentric with the circle $x^2 + y^2 - 6x + 12y + 15 = 0$ and has double of its area.
$[$Hint: Concentric circles have the same centre$]$
View full solution →Find the coordinates of a point on the parabola $y^2 = 8x$ whose focal distance is $4.$
View full solution →Find the equation of a circle whose centre is $(3, -1)$ and which cuts off a chord of length $6$ units on the line $2x - 5y + 18 = 0.$
$[$Hint: To determine the radius of the circle$,$ find the perpendicular distance from the centre to the given line$]$
View full solution →If one end of a diameter of the circle $x^2 + y^2 - 4x - 6y + 11 = 0$ is $(3, 4),$ then find the coordinate of the other end of the diameter.
View full solution →Find the equation of a circle of radius $5$ which is touching another circle $x^2 + y^2 - 2x - 4y - 20 = 0$ at $(5, 5).$
View full solution →