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Find the equation of the locus of the centres of all circles which touch the line x = 2a and cut orthogonally the circle x^2 + y^2 = a^2.

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Find the equation of the locus of the centres of all circles which touch the line x = 2a and cut orthogonally the circle x2 + y2 = a2.

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Let S ≡ x2 + y2 + 2gx + 2fy + c = 0 be the required circle. Now S = 0 touches the line x = 2a implies

S = 0 cuts orthogonally x2 + y2 - a2 = 0 implies

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