Question

The equation of the circle passing through the point \((2a, 0)\) and whose radical axis is \(x = \frac{a}{2}\) with respect to the circle \(x^2 + y^2 = a^2\), will be

• A. \(x^2 + y^2 - 2ax = 0\)
• B. \(x^2 + y^2 + 2ax = 0\)
• C. \(x^2 + y^2 + 2ay = 0\)
• D. \(x^2 + y^2 - 2ay = 0\)

Hint:

Radical axis of two circles is obtained by subtracting their equations.

Answer 1

The Correct Option is A

To find the equation of the circle passing through the point \((2a, 0)\) whose radical axis is \(x = \frac{a}{2}\) with respect to the circle \(x^2 + y^2 = a^2\), follow these steps:

1. Identify the given circle: The given circle is described by the equation \(x^2 + y^2 = a^2\). This circle has a center at \((0, 0)\) and a radius of \(a\).
2. Understand the radical axis: The radical axis is a line depicting the locus of points having equal power concerning two circles. Given that this axis is \(x = \frac{a}{2}\), derive the power of point formula for both circles.
3. Use the power of a point for circle: \(S_1 = x^2 + y^2 - a^2\) and \(S_2 = x^2 + y^2 + Dx + Ey + F\). The radical axis is given by \(S_1 - S_2 = 0\)
4. Substitute \(S_1 - S_2 = x^2 + y^2 - a^2 - (x^2 + y^2 + Dx + Ey + F) = 0\).
5. Simplify to get radical axis: \(-Dx - Ey - F - a^2 = 0\). Given that this equation is equivalent to \(x = \frac{a}{2}\), our task is to match the equation to this line.
6. Interpret the given radical axis \(x = \frac{a}{2}\): From the radical axis equation derived, the term with \(x\) is \(-Dx\), and equating it to the given line, only apply constant terms.
7. From the radical axis, match coefficients: the coefficient of \(x\) gives us \(-D = 1\), therefore \(D = -1\), implying \(D\) must satisfy \(-1\cdot a/2 = -Dx \Rightarrow D = -2a\).
8. Resulting equation for circle \((S_2)\):
   • Ensure the circle passes through \((2a, 0)\).
   • Given: the circle's standard form with this \(D\) is \(x^2 + y^2 - 2ax + f = 0\).
   • Substitute point \((2a, 0)\) into \(x^2 + y^2 - 2ax = 0\):
   • \((2a)^2 + 0^2 - 2a \times 2a = 0 \Rightarrow 4a^2 - 4a^2 = 0\).
   • The condition satisfies this equation, confirming validity.
9. Conclusion: The equation of the circle is \(x^2 + y^2 - 2ax = 0\).