The angle between tangents to the parabola $y^2 = 4ax$ at the points where it intersects with the line $x - y - a = 0$ is

Let $P(\alpha, \beta)$ be a point on the parabola $y^2 = 4x$ which is at minimum distance from the circle $x^2 + y^2 - 4x - 20y + 103 = 0$. Then $\alpha \beta$ is

Let $Z$ be the point of intersection of the axis and the directrix of the parabola $4x^2 - 12x + 4y + 5 = 0$. If $S$ is its focus,then the point which divides $SZ$ in the ratio $2:1$ 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 containing the right angle be $C = (4, 8)$ and the locus of the centroid of $\triangle ABC$ be a conic $C_o$. Then three times the length of the latus rectum of $C_o$ is . . . . . .

For the parabola $y^2 = x$,let $PQ$ be a chord such that one endpoint $P$ is $(4, -2)$ and the chord is perpendicular to the axis of the parabola. What is the slope of the normal at $Q$?

$A$ particle is moving in the $xy$-plane along a curve $C$ passing through the point $(3, 3)$. The tangent to the curve $C$ at the point $P$ meets the $x$-axis at $Q$. If the $y$-axis bisects the segment $PQ$,then $C$ is a parabola with