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(A) Let the point $P$ be $(h, k)$.The equation of the pair of tangents from $P(h, k)$ to the circle $x^2 + y^2 = a^2$ is given by $SS_1 = T^2$.$(x^2 +

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$x^2 + y^2 = a^2 + b^2$

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(A) Let the point $P$ be $(h, k)$.The equation of the pair of tangents from $P(h, k)$ to the circle $x^2 + y^2 = a^2$ is given by $SS_1 = T^2$.$(x^2 + y^2 - a^2)(h^2 + k^2 - a^2) = (xh + yk - a^2)^2$.Expanding this,the coefficient of $x^2$ is $(h^2 + k^2 - a^2) - h^2 = k^2 - a^2$.The coefficient of $y^2$ is $(h^2 + k^2 - a^2) - k^2 = h^2 - a^2$.Since the tangents are perpendicular,the sum of the coefficients of $x^2$ and $y^2$ must be zero.$(k^2 - a^2) + (h^2 - a^2) = 0 \implies h^2 + k^2 = 2a^2$.However,the problem states the tangents to the first circle are perpendicular to the tangents to the second circle. This implies the locus of $P$ is the director circle of the system. For the condition where tangents from $P$ to $x^2 + y^2 = a^2$ are perpendicular to tangents from $P$ to $x^2 + y^2 = b^2$,the locus is $x^2 + y^2 = a^2 + b^2$.

If the tangents drawn from a point $P$ to the circle $x^2 + y^2 = a^2$ are perpendicular to the tangents drawn to the circle $x^2 + y^2 = b^2$,then the locus of $P$ is:

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