A pair of tangents are drawn from a point $P$ to the circle $x^{2} + y^{2} = 1$. If the tangents mak

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A pair of tangents are drawn from a point $P$ to the circle $x^{2} + y^{2} = 1$. If the tangents make an intercept of 2 on the line $x = 1$, then locus of $P$ is:

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[Key Concept: Pair of tangents from $P$: use the chord of contact and the angle relation on the director circle.]Equation of pair of tangents $PA$ and $PB$ is$\begin{matrix} (xh + yk - 1)^{2} = \left( x^{2} + ight.\ & \left. \ y^{2} - 1 ight)\left( h^{2} + k^{2} - 1 ight) \ & \end{matrix}$Put $x = 1,$$$(h - 1)^{2} + 2ky(h - 1) = y^{2}\left( h^{2} - 1 ight)$$$$\begin{matrix} & \ \ & y^{2}(h + 1) - 2ky - (h - 1) = 0 - y_{1} \ & AB = \left| y_{1} - y_{2} ight| = 2 \end{matrix}$$$$\begin{matrix} \Rightarrow \ 4 & \ = \frac{4k^{2}}{(h + 1)^{2}} + \frac{4(h - 1)}{(h + 1)} \ (h + 1) & \ = k^{2} + \left( h^{2} - 1 ight) \Rightarrow k^{2} = 2(h + 1) \ y^{2} & \ = 2(x + 1) \end{matrix}$$

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