List of top Mathematics Questions on Applications of Conics

The eccentricity of an ellipse \(E\) with centre at the origin \(O\) is \( \frac{\sqrt3}{2} \) and its directrices are \( x=\pm \frac{4\sqrt6}{3} \). Let \( H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) be a hyperbola whose eccentricity is equal to the length of semi-major axis of \(E\), and whose length of latus rectum is equal to the length of minor axis of \(E\). Then the distance between the foci of \(H\) is :

Let the eccentricity $e$ of a hyperbola satisfy the equation $6e^2 - 11e + 3 = 0$. If the foci of the hyperbola are $(3, 5)$ and $(3, -4)$, then the length of its latus rectum is :

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola $y^2 = 16x$. Let the vertex B containing the right angle be $(4, 8)$ and the locus of the centroid of $\Delta ABC$ be a conic $C_0$. Then three times the length of latus rectum of $C_0$ is _______.

Let A be the point \( (3, 0) \) and circles with variable diameter AB touch the circle \( x^2 + y^2 = 36 \) internally. Let the curve \( C \) be the locus of the point B. If the eccentricity of \( C \) is \( e \), then \( 72e^2 \) is equal to _______.

If the points of intersection of two distinct conics \(x^2 + y^2 = 4b\) and \(\frac{x^2}{16} + \frac{y^2}{b^2} = 1\) lie on the curve \(y^2 = 3x^2\) then \( 3\sqrt{3} \) times the area of the rectangle formed by the intersection points is __.