List of top Mathematics Questions on Circle and Parabola Geometry asked in JEE Main

Let the directrix of the parabola $P : y^2 = 8x$, cut $x$-axis at the point A. Let $B(\alpha, \beta), \alpha>1$, be a point on P such that the slope of AB is 3/5. If BC is a focal chord of P, then six times the area of $\Delta ABC$ is :

Let the point P be the vertex of the parabola $y = x^2 - 6x + 12$. If a line passing through the point P intersects the circle $x^2 + y^2 - 2x - 4y + 3 = 0$ at the points R and S, then the maximum value of $(PR + PS)^2$ is :

Let the line $x-y=4$ intersect the circle $C:(x-4)^2+(y+3)^2=9$ at the points $Q$ and $R$. If $P(\alpha,\beta)$ is a point on $C$ such that $PQ=PR$, then $(6\alpha+8\beta)^2$ is equal to ______.

Let $x=-9$ be a directrix of an ellipse $E$, whose centre is at the origin and eccentricity is $ \frac13 $. Let $P(\alpha,0), \alpha>0$, be a focus of $E$ and $AB$ be a chord passing through $P$. Then the locus of the mid point of $AB$ is :

Let P be a moving point on the circle $x^2 + y^2-6x-8y + 21 = 0$. Then, the maximum distance of P from the vertex of the parabola $x^2 + 6x + y + 13 = 0$ is equal to:

Let $\alpha = 3 \sin^{-1} \left( \frac{6}{11} \right)$ and $\beta = 3 \cos^{-1} \left( \frac{4}{9} \right)$, where inverse trigonometric functions take only the principal values.
Given below are two statements:
Statement I: $\cos(\alpha + \beta)>0$.
Statement II: $\cos(\alpha) < 0$.
In the light of the above statements, choose the correct answer:

A circle touches the parabola $ y^2 = 9x $ at $ (4, 6) $ and the positive x-axis. Find the radius of the circle.

Let the shortest distance from $ (a, 0) $, where $ a > 0 $, to the parabola $ y^2 = 4x $ be 4. Then the equation of the circle passing through the point $ (a, 0) $ and the focus of the parabola, and having its center on the axis of the parabola is: