Q.20. Area of a triangle formed by chord of a circle and tangents at the end-points of the chord.

Question 20.
(a,b) is the midpoint of the chord AB of the circle x^2 + y^2 = r^2. The tangents at A, B meet at C, then prove that the area of triangle ABC = [(r^2-a^2-b^2)^3/2]/√(a^2+b^2).


Answer 20.
Let O be the centre of the circle perpendicular from which meets AB in P.
ΔOAC is a right triangle and AP is perpendicular to OC.
=> CP*OP = AP^2
=> CP = AP^2/OP
P = (a, b)
=> OP = √(a^2+b^2)
and
AP^2 = OA^2 - OP^2 = r^2 - (a^2 + b^2)
=> AP^3 = (r^2 - a^2 - b^2)^(3/2)
Area of ΔABC
= CP * AP
= (AP^2/OP) * AP
= AP^3 / OP
= (r^2-a^2-b^2)^3/2)/√(a^2+b^2)
[Plugging the values of AP^3 and OP obtained as above.]


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