If one of the diameters of the circle, given by the...

If one of the diameters of the circle, given by the equation, $x^{2}+y^{2}-4 x+6 y-12=0$, is a chord of a circle $S$, whose centre is at $(-3,2)$, then the radius of $S$ is
(A) 10
(B) $5 \sqrt{2}$
(C) $5 \sqrt{3}$
(D) 5

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Key Concepts

Chord of a Circle and Its Properties

A chord is a line segment with both endpoints on the circle, and its perpendicular bisector always passes through the circle’s center. The relationship between a chord’s length, the radius of the circle, and the perpendicular distance from the center to the chord is encapsulated by the formula: chord length = 2?(R² - d²), which is crucial for linking chord properties to the circle's dimensions.

Circle Geometry and Diameters

A diameter is a special chord that passes through the center of the circle, making it the longest possible chord. Recognizing when a line segment serves as a diameter in one circle and potentially a chord in another involves understanding the geometric relationships between the chords, the centers, and the distances involved.

Standard Form of a Circle Equation

Rewriting the general equation of a circle in standard form involves completing the square for both x and y terms. This process reveals the circle's center and radius, which are essential in analyzing the circle's geometry.

Distance Formula

The distance formula in coordinate geometry, derived from the Pythagorean theorem, is key to calculating the distance between two points. This is fundamental in determining distances such as that between circle centers or between a circle's center and a chord.

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