CircleHardBloom L3
Question
If from a moving point $P$ on $x^{2} + y^{2} = 4$ tangents $PA$ and $PB$ are drawn to $x^{2} + y^{2} = a^{2}$, then the locus of the circumcentre of triangle $PAB$, is $(0 < |a| < 2)$ :
Options
A.$x^{2} + y^{2} = a^{2} + 4$
B.$x^{2} + y^{2} = 4 - a^{2}$
C.$x^{2} + y^{2} = 1$
D.$x^{2} + y^{2} = 2$
Solution
[Key Concept: Moving point on $x^{2} + y^{2} = 4$ with tangents to $x^{2} + y^{2} = a^{2}$: the chord of contact's midpoint traces the locus.]
Circumcircle of $\bigtriangleup PAB$has $OP$ as diameter If circumcenter $\equiv (h,k)$
$$\Rightarrow \ P \equiv (2h,2k) $$$P$ lies on $x^{2} + y^{2} = 4$
$$\Rightarrow (2h)^{2} + (2k)^{2} = 4 $$∴ Locus of circumcentre is $x^{2} + y^{2} = 1$.
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