CircleHardBloom L3
Question
The locus of points of intersection of the tangents to $x^{2} + y^{2} = a^{2}$ at the extremeties of a chord of circle $x^{2} + y^{2} = a^{2}$ which touches the circle $x^{2} + y^{2} - 2ax = 0$ is/are:
Options
A.$y^{2} = a(a - 2x)$
B.$x^{2} = a(a - 2y)$
C.$x^{2} + y^{2} = (x - a)^{2}$
D.$x^{2} + y^{2} = (y - a)^{2}$
Solution
Sol. [Key Concept: Tangents at the ends of a chord meet on the polar; for a chord subtending a fixed angle the meeting point traces a concentric circle.]
$PQ$ is chord of contact of $(h,k)$ w.r.t. $x^{2} + y^{2} = a^{2}$
⇒ Equation of $PQ$ is $xh + yk = a^{2}$
$\because\ PQ$ is tangent to $(x - a)^{2} + y^{2} = a^{2}$
$${\Rightarrow \frac{\left| ah - a^{2} \right|}{\sqrt{h^{2} + k^{2}}} = a }{\Rightarrow (h - a)^{2} = h^{2} + k^{2} }$$∴ Locus is $(x - a)^{2} = x^{2} + y^{2}$
i.e., $y^{2} = a(a - 2x)$
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