Find the area of the triangle formed by the tangents from the point (h,k) to the circle x2+y2=a2 and their chord of contact.
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Solution
Equation of the chord of contact AB of tangents drawn from the point P(h,k) to the given circle is hx+ky−a2=0. OM= perp. distance of (0,0) from AB. ∴OM=−a2√(h2+k2) ∴AM2=OA2−OM2=a2−a4h2+k2 =a2(h2+k2−a2)(h2+k2) ∴AB=2AM=2a√(h2+k2−a2h2+k2) PM= perp. distance of (h,k) from AB ∴PM=h2+k2−a2√(h2+k2) △PAB=12PM.AB =12.h2+k2−a2√(h2+k2).2a√(h2+k2−a2h2+k2) =a(h2+k2−a2)3/2h2+k2.....(2).