EXERCISE-4

$P \& Q$ are the points of contact of the tangents drawn from the point $T$ to the parabola $y^2=4ax$. If $PQ$ be the normal to the parabola at $P$ then prove that $TP$ is bisected by the directrix.

The normal at a point $P$ to the parabola $y^2=4ax$ meets its axis at $G$. $Q$ is another point on the parabola such that $QG$ is perpendicular to the axis of the parabola. Prove that $Q{G}^2-P{G}^2=$ constant.

Three normals to $y^2=4x$ pass through the point $\left(15, 12\right).$ Show that one of the normals is given by $y=x-3$ $\&$  find the equations of the others.

Find the condition on$\text{'} a\text{'} \& b\text{'}$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by.$

Let $S$ is the focus of the parabola $y^2=4ax$ and $X$ the foot of the directrix, $PP\text{'}$ is a double ordinate of the curve and $PX$ meets the curve again in $Q.$ Prove that $P\text{'}Q$ passes through focus.

If $\left(x_1, y_1\right), \left(x_2, y_2\right)$ and $\left(x_3, y_3\right)$ be three points on the parabola $y^2=4ax$ and the normals at these points meet in a point, then prove that $\frac{x_1-x_2}{y_3}+\frac{x_2-x_3}{y_1}+\frac{x_3-x_1}{y_2}=0$

A variable chord joining points $P\left(t_1\right)$ and $Q\left(t_2\right)$ of the parabola $y^2=4ax$ subtends a right angle at a fixed point $t_0$ of the curve. Show that it passes through a fixed point. Also find the co-ordinates of the fixed point.

Show that a circle circumscribing the triangle formed by three co-normal points passes through the vertex of the parabola and its equation is, $2\left(x^2+y^2\right)-2(h+2a)x-ky=0,$ where $(h, k)$ is the point from where three concurrent normals are drawn.