Question

If the chord of contact of tangents from a point $P$ to the parabola $y^2=4ax$ touches the parabola $x^2=4by,$ then the locus of $P$ is

• A. a circle
• B. a parabola
• C. a straight line
• D. none of these

Answer 1

The correct option is B a parabola

Let $P(x_1,y_1)$ be a point.

$\therefore$ Chord of contact is ${yy}_1=2a(x+x_1)$ i.e. $x=\frac{y_1}{2a}y-x_1$

$x=my+c,$ where $y=\frac{y_1}{2a},c=-x_1$

$\therefore$ Now, $x=mx+c$ touches $x^2=4by$ if $c=\frac{b}{m}$

i.e. if $-x_1=\frac{b}{\frac{y_1}{2a}}\Rightarrow -x_1{y}_1=2ab$

$\therefore$ Locus of $(x_1,y_2)$ is $-xy=2ab$

i.e. $x=-2ab$ which is a hyperbola.