Question

A circle of radius 5 is tangent to the line 4x - 3y = 18 at M(3,-2) and lies above the line. The equation of the circle, is-

• A. x² + y² - 6x + 4y - 12 = 0
• B. x² + y² + 2x - 2y - 3 = 0
• C. x² + y² + 2x - 2y - 23 = 0
• D. x² + y² + 6x + 4y - 12 = 0

Answer 1

Answer: (C)

x² + y² + 2x - 2y - 23 = 0

Answer 2

The center of the circle lies on the normal to the tangent line at the point of tangency. The distance from the center to the point of tangency is the radius. Solving these conditions yields two possible centers. The condition that the circle lies "above the line" $4x-3y=18$ (which is equivalent to $4x-3y < 18$) selects the correct center. The equation of the circle is then formed using the center and radius.