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Question

$Consider the locus of centre of circle which touches circle x^{2} + y^{2} = 4 and line x = 4 . The distance of the vertex of$
$the locus from origin is$

Moderate

Solution

$Radius of variable circle is 4 - h .$
$It touches x^{2} + y^{2} = 4$
$\begin{aligned} Then 2 + 4 - h = \sqrt{h^{2} + k^{2}} \\ or x^{2} + y^{2} = x^{2} - 12 x + 36 \end{aligned}$
$\begin{array}{l} \Rightarrow y^{2} = - 12 (x - 3) \\ The vertex (3, 0) . \end{array}$

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