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(D) Let the given point be $P(4, -3)$ and the given circle be $x^2 + y^2 + 4x - 10y - 7 = 0$. The center of the circle $C$ is $(-2, 5)$ and the radius

Vedclass · Accepted Answer

$20$

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(D) Let the given point be $P(4, -3)$ and the given circle be $x^2 + y^2 + 4x - 10y - 7 = 0$. The center of the circle $C$ is $(-2, 5)$ and the radius $r$ is $\sqrt{(-2)^2 + (5)^2 - (-7)} = \sqrt{4 + 25 + 7} = \sqrt{36} = 6$. The distance between the point $P$ and the center $C$ is $CP = \sqrt{(4 - (-2))^2 + (-3 - 5)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. The maximum distance from the point to the circle is $d_{max} = CP + r = 10 + 6 = 16$. The minimum distance from the point to the circle is $d_{min} = |CP - r| = |10 - 6| = 4$. The sum of the minimum and maximum distance is $d_{max} + d_{min} = 16 + 4 = 20$.

The sum of the minimum and maximum distance of the point $(4, -3)$ to the circle $x^2 + y^2 + 4x - 10y - 7 = 0$ is:

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