Two chords of length 5 are drawn from any point (3,4) on the circle $4x^2 \, + \, 4y^2 \, - \, 24x \, - \, 7y \, = \, 0$ then their equations are given by $y - \, 4 \, = \, \pm \, \dfrac{4}{3} \, (x \, - \, 3)$

The center of the given circle is found by completing the square, which yields coordinates (3, 7/8). The slope of the line connecting this center to the point (3, 4) is evaluated as (4 - 7/8) divided by (3 - 3), which is undefined, indicating a vertical line. The length of the chord is calculated using the perpendicular distance from the center to the chord line formula, yielding the square root of 25, which equals 5 as required. Since the given slopes of 4/3 and -4/3 both produce this exact perpendicular distance and chord length of 5, the stated equations are true.

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