Prove that the circle x 2 + y 2 + 2ax + c 2 = 0 and x 2 + y…

Maths Circle and System of Circles Family of Circles, Locus Subjective Type

Published on: August 13, 2026

Prove that the circle x 2 + y 2 + 2ax + c 2 = 0 and x 2 + y 2 + 2by + c 2 = 0 touches each other

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The correct answer is:

CHECK THE SOLUTION.

Subtract to get common tangent and drop pependicular from centre on any one circle and equate it to its radius.

Radical axes is ax – by = 0 which touches both the circle

Now x 2 + + 2ax + c 2 = 0

⇒ (b 2 + a 2 )x 2 + 2ab 2 x + b 2 c 2 = 0

⇒ 4a 2 b 2 – 4(b 2 + a 2 ) b 2 c 2 = 0

⇒ a 2 b 2 = c 2 (b 2 + a 2 )

⇒ = +

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