Question

$A$ line passing through the point $P(2,3)$ meets the lines represented by $x^2-2xy-y^2=0$ at the point $A$ and $B$ such that $PA.PB=17$, the equation of the line is

• A. $x=2$
• B. $y=3$
• C. $3x-2y=0$
• D. $x=5$

Answer 1

Answer: (B)

$y=3$

Any point on the line $=(2+r\cos \theta ,3+r\sin \theta )$
Let $PA=r_1$ and $PB=r_2$
where $r_1,r_2$ are roots of the equation
$r^2(\cos 2\theta -\sin 2\theta )-2r(\cos \theta +5\sin \theta )-17=0$
As $PA.PB=17$
And $r_1{r}_2=\frac{17}{\cos 2\theta -\sin 2\theta }=17$
$\Rightarrow \cos 2\theta -\sin 2\theta =1\Rightarrow \theta =0^o$
Hence $y=3$ is equation