Question

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

• A. straight line
• B. circle
• C. parabola
• D. ellipse

Answer 1

The correct option is A straight line

Circle TPQ is S+λP=0 where P is chord of contact of T(h,k).

Circumcircle is $(x^2+y^2-a^2)+\lambda (hx+ky-a^2)$

It passes through T(h,k)

$\therefore (h^2+k^2-a^2)=\lambda (h^2+k^2-a^2)=0$

or λ=−1

Circle is$x^2+y^2-hx-ky=0$

Its centre $x=\frac{h}{2},y=\frac{k}{2}$

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But (h,k) lies on

px+qy+r=0,

∴ph+qk+r=0.

Eliminating h,k we get the locus as

$px+qy+r=0$.

Hence we get straight line