Dr. SK Goyal Solutions for Chapter: Parabola, Exercise 4: EXERCISE ON LEVEL-I

Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$.

On the parabola $y^2=4ax$ three point $P,Q,R$ are taken so that their ordinates are in geometrical progression. Prove that the tangents at $P$ and $R$ meet on abscissa of $Q$.

Prove that three normals can be drawn from the point $(c, 0)$ to the parabola $y^2=x$if $c>\frac{1}{2}$ and then one of the normals is always the axis of the parabola

Show that the length of the tangent to the parabola $y^2=4ax$ is intercepted between its point of contact and the axis of the parabola is bisected by the tangent at the vertex

Find the centre and radius of the smaller of the two circles that touch the parabola $75y^2=64\left(5x-3\right)$ at $\left(6/5,8/5\right)$ and the $x$-axis

The normals at $P,Q$ the ends of a focal chord of a parabola meets the parabola again in $P^{\text{'}}$ and $Q^{\text{'}}$ respectively. Show that $PQ$ is parallel to $P^{\text{'}}{Q}^{\text{'}}$ and that $3PQ=P^{\text{'}}{Q}^{\text{'}}$.

If perpendiculars be drawn from two fixed points on the axis of a parabola equidistant from the focus, on any tangent to it, show that the difference of their squares is constant.

Find the condition that the chord $t_1{t}_2$ of the parabola $y^2=4ax$ passes through the point $\left(a,3a\right)$. Find the locus of intersection of the tangents at $t_1$ and $t_2$ under this condition.