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(A) The equation of the circle is $x^2 + y^2 - 2\alpha x + 6y + \alpha^2 - 16 = 0$. Comparing with $x^2 + y^2 + 2gx + 2fy + c = 0$,we get $g = -\alpha

Vedclass · Accepted Answer

$16 + 4\sqrt{6}$

Vedclass · Answer

(A) The equation of the circle is $x^2 + y^2 - 2\alpha x + 6y + \alpha^2 - 16 = 0$. Comparing with $x^2 + y^2 + 2gx + 2fy + c = 0$,we get $g = -\alpha, f = 3, c = \alpha^2 - 16$. The power of a point $(x_1, y_1)$ with respect to a circle $S = 0$ is $S(x_1, y_1) = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$. Given power is $9$ at $(4, 2)$: $4^2 + 2^2 - 2\alpha(4) + 6(2) + \alpha^2 - 16 = 9$ $16 + 4 - 8\alpha + 12 + \alpha^2 - 16 = 9$ $\alpha^2 - 8\alpha + 7 = 0$ $\Rightarrow (\alpha - 1)(\alpha - 7) = 0$ $\Rightarrow \alpha = 1, 7$. Case $1$: $\alpha = 1$,circle is $x^2 + y^2 - 2x + 6y - 15 = 0$. $x$-intercept: set $y = 0$ $\Rightarrow x^2 - 2x - 15 = 0$ $\Rightarrow (x-5)(x+3) = 0$ $\Rightarrow x = 5, -3$. Length $= |5 - (-3)| = 8$. $y$-intercept: set $x = 0$ $\Rightarrow y^2 + 6y - 15 = 0$ $\Rightarrow y = -3 \pm \sqrt{9 + 15} = -3 \pm 2\sqrt{6}$. Length $= |(-3 + 2\sqrt{6}) - (-3 - 2\sqrt{6})| = 4\sqrt{6}$. Case $2$: $\alpha = 7$,circle is $x^2 + y^2 - 14x + 6y + 33 = 0$. $x$-intercept: set $y = 0$ $\Rightarrow x^2 - 14x + 33 = 0$ $\Rightarrow (x-11)(x-3) = 0$ $\Rightarrow x = 11, 3$. Length $= |11 - 3| = 8$. $y$-intercept: set $x = 0 \Rightarrow y^2 + 6y + 33 = 0$. Discriminant $D = 36 - 4(33) Total sum of lengths $= 8 + 4\sqrt{6} + 8 = 16 + 4\sqrt{6}$.

If the power of a point $(4, 2)$ with respect to the circle $x^2 + y^2 - 2\alpha x + 6y + \alpha^2 - 16 = 0$ is $9$,then the sum of the lengths of all possible intercepts made by such circles on the coordinate axes is:

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