Question

Tangent are drawn from $P$ to $y^2=4ax.$ If the difference of the ordinates of the point of contact of the two tangents is $l$, then the equation of the locus of $P$ is

• A. $y^2-4ax=l^2$
• B. $y^2-4ax=\frac{l^2}{2}$
• C. $y^2-4ax=\frac{l^2}{4}$
• D. $y^2-4ax=2l^2$

Answer 1

The correct option is C $y^2-4ax=\frac{l^2}{4}$

Given parabola is $y^2=4ax$
Let $P$ be $(h,k)$
Equation of the chord of contact of $P$ with respect to $y^2=4ax$ is
$T=0\Rightarrow yk=2a(x+h)\Rightarrow \frac{yk}{2a}-h=x\cdots \left(1\right)$

Solving equation $\left(1\right)$ with $y^2=4ax$, we get
$y^2=4a(\frac{yk}{2a}-h)\Rightarrow y^2-2ky+4ah=0$
$\therefore$ The sum of the roots is
$y_1+y_2=2k$
Also, the product of the roots is
$y_1\cdot y_2=4ah$

Now, given condition is
$|y_1-y_2|=l\Rightarrow (y_1-y_2{)}^2=l^2\Rightarrow (y_1+y_2{)}^2-4y_1\cdot y_2=l^2\Rightarrow 4k^2-16ah=l^2$
Hence, the locus is
$\Rightarrow y^2-4ax=\frac{l^2}{4}$