Question

A tangent to $y^2=4ax$ meets $x$-axis at $T$ and tangent at vertex $A$ in $P$ and the rectangle $TAPQ$ is completed. Then the locus of $Q$ is given by,

• A. $y^2+4ax=0$
• B. $y^2+2ax=0$
• C. $y^2=2ax$
• D. $y^2+ax=0$

Answer 1

The correct option is D $y^2+ax=0$

Let$Q$ be $(h,k)$

$T(h,0)$ and $P(0,k)$

Let the tangent be, $y=mx+\frac{a}{m}$

$\frac{a}{m}=-hm\Rightarrow m^2=\frac{-a}{h}$...(1)

$\frac{a}{m}=k\Rightarrow m=\frac{a}{k}$...(2)

From (1) and (2)

$\frac{-a}{h}=\frac{a^2}{k^2}$

$\Rightarrow a=\frac{-k^2}{h}$

$ax=-y^2$

$y^2+ax=0$