The locus of the centre of a circle passing t | vedclass

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(A) Let the equation of the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$.Since it cuts the circle $x^2 + y^2 = p^2$ orthogonally,the condition $2g_1g_2 +

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$2ax + 2by - (a^2 + b^2 + p^2) = 0$

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(A) Let the equation of the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$.Since it cuts the circle $x^2 + y^2 = p^2$ orthogonally,the condition $2g_1g_2 + 2f_1f_2 = c_1 + c_2$ gives $2g(0) + 2f(0) = c - p^2$,which implies $c = p^2$.Since the circle passes through $(a, b)$,we have $a^2 + b^2 + 2ga + 2fb + p^2 = 0$.The centre of the circle is $(-g, -f)$. Let the centre be $(x, y)$,so $g = -x$ and $f = -y$.Substituting these into the equation,we get $a^2 + b^2 + 2(-x)a + 2(-y)b + p^2 = 0$.This simplifies to $2ax + 2by - (a^2 + b^2 + p^2) = 0$.

The locus of the centre of a circle passing through $(a, b)$ and cutting orthogonally to the circle $x^2 + y^2 = p^2$ is

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