AP EAMCET · Maths · Circle

\((a, 0)\) and \((b, 0)\) are centres of two circles belonging to a coaxial system of which \(y\)-axis is the radical axis. If radius of one of the circles is ' \(r\) ', then the radius of the other circle is

A \(\left(r^2+b^2+a^2\right)^{1 / 2}\)
B \(\left(r^2+b^2-a^2\right)^{1 / 2}\)
C \(\left(r^2+b^2-a^2\right)^{1 / 3}\)
D \(\left(r^2+b^2+a^2\right)^{1 / 3}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\left(r^2+b^2-a^2\right)^{1 / 2}\)

Step-by-step Solution

Detailed explanation

Let the equation of circle whose centre \((a, 0)\) and radius \((r)\) is \[ \begin{gathered} (x-a)^2+(y-0)^2=r^2 \\ \Rightarrow \quad S_1 \equiv x^2+a^2-2 a x+y^2-r^2=0 \end{gathered} \] and the equation of circle whose centre \((b, 0)\) and radius \(R\) is…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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