Assertion: If the angle between the circles $x^{2} + y^{2} - 12 x - 6 y + 41 = 0$ and $x^{2} + y^{2} + k x + 6 y - 59 = 0$ is $\frac{π}{4}$ then $k = 4$
Reason: If the angle between the circles $S_{1} = 0$ and $S_{2} = 0$ is $θ$ , then $\cos θ = \frac{d^{2} - r_{1}^{2} - r_{2}^{2}}{2 r_{1} r_{2}}$ where $r_{1} & r_{2}$ are radii of circles $S_{1} = 0$ , $S_{2} = 0$ respectively and $d = c_{1} c_{2}$

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(A) is false and (R) is true.

(A) is true and (R) is false.

Both (A) and (R) are true but (R) is not the correct explanation of (A).

Both (A) and (R) are true, and (R) is the correct explanation of (A).

answer is A.

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Detailed Solution

$c_{1} = (6, 3), r_{1} = 2 & c_{2} = (\frac{- k}{2}, - 2), r_{2} = \sqrt{\frac{k^{2}}{4} + 68}$
$\Rightarrow d = \sqrt{{(6 + \frac{k}{2})}^{2} + 36}$
Now, $\cos \frac{π}{4} = \frac{{(6 + \frac{k}{2})}^{2} + 36 - 4 - \frac{k^{2}}{4} - 68}{2.2 \sqrt{\frac{k^{2}}{4} + 68}}$
$\Rightarrow \frac{1}{\sqrt{2}} = \frac{6 k}{4 \sqrt{\frac{k^{2}}{4} + 68}}$
$\Rightarrow k^{2} = 16 \Rightarrow k = 4$