Exercise 3(b)

Two parabolas have the same vertex and equal length of latus rectum such that their axes are at right angle. Prove that the common tangents touch each at the end of latus rectum.

Show that the foot of the perpendicular from focus to the tangent of the parabola $y^2=4ax$ lies on the tangent at vertex.

Show that the tangent at one extremity of a focal chord of a parabola is parallel to the normal at other extremity.

The normal at a point $t_1$ on $y^2=4ax$ meets the parabola again in the point $t_2$. Then prove that $t_1{t}_2+{t_1}^2+2=0$

From an external point $P$ tangents are drawn to the parabola $y^2=4ax$ and these tangents make angles ${\theta }_1,{\theta }_2$ with its axis, such that $\cot {\theta }_1+\cot {\theta }_2$ is a constant $\text{'}d\text{'}$. Then show that all such $P$ lie on a horizontal line.

Show that the common tangents to the circle $2x^2+2y^2=a^2$ and the parabola $y^2=4ax$ intersect at the focus of the parabola $y^2=-4ax$.

The sum of the ordinates of two points on $y^2=4ax$ is equal to the sum of the ordinates of two other points on the same curve. Show that the chord joining the first two points is parallel to the chord joining the other two points.

If a normal chord at point $\text{'}t\text{'}$ on the parabola $y^2=4ax$ subtends a right angle at vertex, then prove that $t=\pm \sqrt{2}$.