Question

Tangents $TP$ and $TO$ are drawn from a point $T$ to the circle $x^2+y^2=a^2$.If the point $T$ lies on the line $px+qy=r$, then the locus of centre of the circumcircle of $\Delta TPO$ is

• A. $px+qy=\frac{r}{3}$
• B. $px+qy=\frac{r}{2}$
• C. $px+qy=2r$
• D. $px+qy=r$

Answer 1

Answer: (B)

$px+qy=\frac{r}{2}$

Let $T=(h,k)$
$\therefore ph+qk=r$ ---(1)
Equation of $OT$ is
$y=\frac{k}{h}x$
So, circumcircle of triangle $PTO$ lie on line $OT$
As we know, $C(0,0),P,O$ and $T(h,k)$ are con-cyclic points

So, center of circle passing through $P,T,$ and $O$ is midpoint of $OT$ because $OT$ is diameter of circle.
So, centre = $(\frac{h}{2},\frac{k}{2})$
So, $x=\frac{h}{2}$ and $y=\frac{k}{2}$
Given $(h,k)$ lies on $px+qy=r$ of $ph+qk=r$
So, locus of of centre is $2px+2qy=r$
$\Rightarrow px+qy=\frac{r}{2}$