Question

A circle has its center at C(-3,-2) and radius square root of 76. Find the length of its chord bisected at M(4,-1).

Answer 1

Answer on Question #56369 – Math – Analytic Geometry

Question

A circle has its center at C(-3,-2) and radius square root of 76. Find the length of its chord bisected at M(4,-1).

Solution

A radius drawn to the midpoint of a chord is perpendicular to the chord (Two points that are equidistant from the endpoints of a segment lie on the perpendicular bisector of the segment).

We find $CM$ using the distance formula:

$$CM = \sqrt{(4 + 3)^2 + (-1 + 2)^2} = \sqrt{50} = 5\sqrt{2}.$$

$\Delta CMA$ is a right triangle. $CA$ is a radius so $CA = \sqrt{76}$.

Using the Pythagorean Theorem, we find $AM$:

$$AM^2 + CM^2 = CA^2;$$ $$AM^2 = CA^2 - CM^2;$$ $$AM^2 = 76 - 50 = 26;$$ $$AM = \sqrt{26}.$$

Since $M$ is a midpoint of $AB$, we have $AB = 2\sqrt{26}$.

Answer: the length of the chord bisected at (4,-1) of the given circle is $2\sqrt{26}$.

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