Q.291. Equation of a circle with endpoints of the chord subtending angle a at the circumference

Question 291.
Find the equation of a circle having points (x1, y1) and (x2, y2) as the end-points of a chord  which subtends an angle "a" at the circumference.

Answer 291.
Let P (x, y) be any point on the circle containing chord AB , where
A = (x1, y1) and B = (x2, y2) and ∠APB = a or π - a
 Slope of AP = (y-y1) / (x-x1) = m1
Slope of BP = (y-y2) / (x-x2) = m2

tan (∠APB) = tana or tan(π-a) = (m2 - m1) / (1 + m1m2)
=> 1 + m1m2 ± cota (m2 - m1) = 0
=> 1 + [(y-y1)(y-y2)] / [(x-x1)(x-x2)] ± cota [(y-y2) / (x-x2) - (y-y1) / (x-x1)] = 0
=> (x-x1)(x-x2) + (y-y1)(y-y2) ± cota [(x-x1)(y-y2) - (x-x2)(y-y1)] = 0
which is the equation of the circle.

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