Assertion :The equation of the circle passing through the point $(2 a, 0)$ and whose radical axis is $x = \frac{a}{2}$ with respect to the circle $x^{2} + y^{2} = a^{2}$ , will be $x^{2} + y^{2} - 2 a x = 0$ Reason: The equation of radical axis of two circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0$ and $x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ is $2 (g_{1} - g_{2}) x + 2 (f_{1} - f_{2}) y + (c_{1} - c_{2}) = 0$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Both Assertion and Reason are incorrect

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Let S1 = $x^{2} + y^{2} - 2 a x = 0$

Let S2 = $x^{2} + y^{2} - a = 0$

Equation of radical axis is given by

$S 1 - S 2 = - 2 a x + a^{2} = 0$

$⟹ 2 a x = a^{2}$

$⟹ x = \frac{a}{2}$

$(2 a, 0)$ must pass through S1

$⟹ (2 a)^{2} + 0 - 2 a (2 a) = 0$

$0 = 0$

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x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0

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x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0

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∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + y^{2} & - x & - y c_{1} & g_{1} & f_{1} c_{2} & g_{2} & f_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

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Assertion :The equation of the circle passing through the point (2a,0) and whose radical axis is x=a2 with respect to the circle x2+y2=a2, will be x2+y2−2ax=0 Reason: The equation of radical axis of two circles x2+y2+2g1x+2f1y+c1=0 and x2+y2+2g2x+2f2y+c2=0 is 2(g1−g2)x+2(f1−f2)y+(c1−c2)=0

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for AssertionLet S1 = x2+y2–2ax=0 Let S2 = x2+y2–a=0 Equation of radical axis is given by S1–S2=−2ax+a2=0 ⟹2ax=a2 ⟹x=a2 (2a,0) must pass through S1 ⟹(2a)2+0−2a(2a)=0 0=0