List of top Mathematics Questions on Circle and Parabola Geometry

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:

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 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 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 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:
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:
A circle with center at \( (x, y) = (0.5, 0) \) and radius = 0.5 intersects with another circle with center at \( (x, y) = (1, 1) \) and radius = 1 at two points. One of the points of intersection \( (x, y) \) is: