Dr. SK Goyal Solutions for Chapter: Parabola, Exercise 2: INTRODUCTORY EXERCISE 5.2

The circle $x^2+y^2+4\lambda x=0$ which $\lambda \in R$ touches the parabola $y^2=8x$. The value of $\lambda$ is given by:

The set of points on the axis of the parabola $y^2-4x-2y+5=0$ from which all the three normals to the parabola are real is :

Prove that any three tangents to a parabola whose slopes are in harmonic progression enclose a triangle of constant area.

A chord of parabola $y^2=4ax$ subtends a right angle at the vertex. Find the locus of the point of intersection of tangents at its extremities.

Find the equation of the normal to the parabola $y^2=4x$ which is :
(a) parallel to the line $y=2x-5$
(b) Perpendicular to the line $2x+6y+5=0$

The ordinates of points $P$ and $Q$ on the parabola $y^2=12x$ are in the ratio $1:2$. Find the locus of the point of intersection of the normals to the parabola at $P$ and $Q$.

The normals at $P,Q,R$ on the parabola $y^2=4ax$ meet in a point on the line $y=c$. Prove that the sides of the triangle $PQR$ touch the parabola $x^2=2cy$.

The normals are drawn from $\left(2\lambda ,0\right)$ to the parabola $y^2=4x$. Show that $\lambda$ must be greater than $1$. One normal is always the $x$-axis. Find $\lambda$ for which the other two normals are perpendicular to each other.