List of top Mathematics Questions on Conic sections

Let the ellipse $ 3x^2 + py^2 = 4 $ pass through the centre $ C $ of the circle $ x^2 + y^2 - 2x - 4y - 11 = 0 $ of radius $ r $. Let $ f_1, f_2 $ be the focal distances of the point $ C $ on the ellipse. Then $ 6f_1 f_2 - r $ is equal to

Let $$ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b \quad \text{and} \quad H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1. $$ Let the distance between the foci of $E$ and the foci of $H$ be $2\sqrt{3}$. If $a - A = 2$, and the ratio of the eccentricities of $E$ and $H$ is $\frac{1}{3}$, then the sum of the lengths of their latus rectums is equal to:

Consider the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, $ having one of its foci at $ P(-3, 0) $. If the latus rectum through its other focus subtends a right angle at $ P $, and $ a^2b^2 = \alpha\sqrt{2} - \beta, \quad \alpha, \beta \in \mathbb{N}, $ then find $ \alpha $ and $ \beta $.

If $ S $ and $ S' $ are the foci of the ellipse \[ \frac{x^2}{18} + \frac{y^2}{9} = 1 \] and $ P $ is a point on the ellipse, then \[ \min (SP \cdot S'P) + \max (SP \cdot S'P) \] is equal to:

Let one focus of the hyperbola $ H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $ and the corresponding directrix be $ x = \dfrac{9}{\sqrt{10}} $. If $ e $ and $ l $ respectively are the eccentricity and the length of the latus rectum of $ H $, then $ 9 \left(e^2 + l \right) $ is equal to:

If a normal is drawn at a variable point P(x, y) on the curve $9x^2+16y^2-144=0$, then the maximum distance from the centre of the curve to the normal is

If $\theta$ is the acute angle between the tangents drawn from the point (1,5) to the parabola $y^2 = 9x$ then

Let P be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and let the perpendicular drawn through P to the major axis meet its auxiliary circle at Q. If the normals drawn at P and Q to the ellipse and the auxiliary circle respectively meet in R, then the equation of the locus of R is

If the tangent drawn at the point $P(3\sqrt{2}, 4)$ on the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1$ meets its directrix at $Q(\alpha, \beta)$ in the fourth quadrant then $\beta = $

The midpoint of the chord of the ellipse $x^2+\frac{y^2}{4}=1$ formed on the line $y=x+1$ is

If the normals drawn at the points $P\left(\frac{3}{4}, \frac{3}{2}\right)$ and $Q(3,3)$ on the parabola $y^2 = 3x$ intersect again on $y^2=3x$ at R, then R =

The equation of the locus of a point which is at a distance of 5 units from a fixed point (1,4) and also from a fixed line 2x+3y-1=0 is

If $ x - y - 3 = 0 $ is a normal drawn through the point $ (5, 2) $ to the parabola $ y^2 = 4x $, then the slope of the other normal that can be drawn through the same point to the parabola is?

The roots of the quadratic equation $ x^2 - 16 = 0 $ are:

The tangent drawn at an extremity (in the first quadrant) of latus rectum of the hyperbola $$ \frac{x^2}{4} - \frac{y^2}{5} = 1 $$ meets the x-axis and y-axis at $ A $ and $ B $ respectively. If $ O $ is the origin, find $$ (OA)^2 - (OB)^2. $$

If $ \theta $ is the angle subtended by a latus rectum at the center of the hyperbola having eccentricity $$ \frac{2}{\sqrt{7} - \sqrt{3}}, $$ then find $ \sin \theta $.

If the normal drawn at the point $$ P \left(\frac{\pi}{4}\right) $$ on the ellipse $$ x^2 + 4y^2 - 4 = 0 $$ meets the ellipse again at $ Q(\alpha, \beta) $, then find $ \alpha $.

Let $ P(x_1, y_1) $ and $ Q(x_2, y_2) $ be two distinct points on the ellipse $$ \frac{x^2}{9} + \frac{y^2}{4} = 1 $$ such that $ y_1 > 0 $, and $ y_2 > 0 $. Let $ C $ denote the circle $ x^2 + y^2 = 9 $, and $ M $ be the point $ (3, 0) $. Suppose the line $ x = x_1 $ intersects $ C $ at $ R $, and the line $ x = x_2 $ intersects $ C $ at $ S $, such that the $ y $-coordinates of $ R $ and $ S $ are positive. Let $ \angle ROM = \frac{\pi}{6} $ and $ \angle SOM = \frac{\pi}{3} $, where $ O $ denotes the origin $ (0, 0) $. Let $ |XY| $ denote the length of the line segment $ XY $. Then which of the following statements is (are) TRUE?

The roots of quadratic equation $ 2x^2 + x - 4 = 0 $ are:

The zeroes of the quadratic polynomial $ x^2 + x - 2 $ are:

If $ \alpha, \beta $ are the roots of $ 2x^2 - 4x + 5 = 0 $, then $ (\alpha + 1)(\beta + 1) = $:

Axis of parabola is $ x = 0 $. If the vertex is at a distance of 3 from the origin above the x-axis, find the vertex.

The maximum number of common normals of $ y^2 = 4ax $ and $ x^2 = 4by $ is equal to:

Equation of parabola having focii (-3, 1) and (3, 1)

The length of the latus rectum of $ x^2 + 3y^2 = 12 $ is: