Let the equation of the tangent at a point $P$ on the parabola $x^2-4x-4y+16=0$ be $2x-y-5=0$. If the equation of the normal drawn at $P$ to this parabola is $ax+y+c=0$,then find the value of $ac$.

From the point $(-1, 2)$, tangents are drawn to the parabola $y^2 = 4x$. The area of the triangle formed by the chord of contact and the tangents is: (in $\sqrt{2}$)

The slope of a chord of the parabola $y^2 = 4ax$ which is normal at one end and which subtends a right angle at the origin is

If $m_1$ and $m_2$ are the slopes of the tangents drawn from the point $(2, 3)$ to the parabola $y^2 = 4x$,then what is the value of $\frac{1}{m_1} + \frac{1}{m_2}$?

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 . . . . . .

The equation of the parabola with its vertex at the origin,axis on the $y$-axis and passing through the point $(6, -3)$ is