Question

If $b,k$ are intercepts of a focal chord of the parabola $y^2=4ax$ then $k$ is equal to

• A. $\frac{ab}{b-a}$
• B. $\frac{b}{b-a}$
• C. $\frac{a}{b-a}$
• D. $\frac{ab}{a-b}$

Answer 1

Answer: (A)

If coordinates of one end of focal chord are $P\left(a{t}^2,2at\right)$ then the coordinates of other end will be $Q\left(\frac{a}{t^2},\frac{-2a}{t}\right)$.
$\therefore SP=\sqrt{{\left(a{t}^2-a\right)}^2+(2at-0{)}^2}$
$=a\left(t^2+1\right)=b$ (given) .... (1)
and, $SQ=\sqrt{{\left(\frac{a}{t^2}-a\right)}^2+{\left(\frac{-2a}{t}-0\right)}^2}$
$=\sqrt{a\left(\frac{1}{t^2}+1\right)}=k$ (given) ....(2)

Dividing (1) by (2), we get $t^2=\frac{b}{k}$.
Putting in (1), $\frac{b}{k}+1=\frac{b}{a}$
$\Rightarrow k=\frac{ab}{b-a}$.