Consider a circle C which touches the y - axis at (0, 6) and cuts...

We have, a circle $C$ which touches the $y -$ axis at \left(0, 6\right)$and cuts off an intercept$6\sqrt{5}$on the$x -$axis.</p><p>We know that, length of the chord of the circle using perpendicular distance from the center is$2\sqrt{{r}^{2} - {d}^{2}}$where$r$is the radius of the circle and$d$is the perpendicular distance from the center to the chord.</p><p>From the given data, we have the following diagram</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/9bbdc738-5a2f-4cc2-b49c-1fea0442b4d4-image.png" style="width: 370px; height: 350px;" /></p><p>Hence, length of the chord of the circle$⇒ 2\sqrt{{r}^{2} - {d}^{2}} = 6\sqrt{5}$</p><p>$⇒ \sqrt{{r}^{2} - {d}^{2}} = 3\sqrt{5}$</p><p>On squaring on both sides</p><p>$⇒ {r}^{2} - {d}^{2} = 45$</p><p>$⇒ {d}^{2} = {r}^{2} - 45$</p><p>Given that circle touches the$y -$axis at \left(0, 6\right)$. Therefore, $d = 6$.

$⇒ {r}^{2} = 45 - 36$

$r = \sqrt{36 + 45}$

$r = 9$.