The number of normals that can be drawn through the point $(9,6)$ to the parabola $y^2=4x$ is

The equation of the line passing through the point $(1/2, 2)$ and tangent to the parabola $y = -\frac{x^2}{2} + 2$ and secant to the curve $y = \sqrt{4 - x^2}$ is

Find the equation of the parabola with vertex $(-3, 0)$ and directrix $x + 5 = 0$.

Variable straight lines $y=mx+c$ make intercepts on the curve $y^2-4ax=0$ which subtend a right angle at the origin. Then the point of concurrence of these lines $y=mx+c$ is

$A$ normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0, -\alpha)$ to the parabola $x^2 = -4ay$,where $a > 0$. Let $L$ be the line passing through $(0, -\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r : s = 1 : 16$,then the value of $24a$ is. . . .

If a normal is drawn to the parabola $y^2 = 4ax$ at the point $(2a, 2\sqrt{2}a)$,what is the length of the normal chord?