If a line drawn from a fixed point M a , b cuts the circle x 2 + y 2 = k 2 at C and D , then M C × M D is equal to

Question

If a line drawn from a fixed point M a , b cuts the circle x 2 + y 2 = k 2 at C and D , then M C × M D is equal to, a 2 + b 2 + k 2, a 2 + b 2 - k 2, a 2 - b 2 - k 2, x 2

MARKS App · Accepted Answer

Let the line have slope tan θ x - a cos θ = y - b sin θ = r x = a + r cos θ , y = b + r sin θ ⇒ x 2 + y 2 = k 2 ⇒ a + r cos θ 2 + ( b + r sin θ ) 2 = k 2 r 2 + 2 r ( a cos θ + b sin θ ) + a 2 + b 2 - k 2 = 0 It has two roots r 1 = M C and r 2 = M D r 1 · r 2 = a 2 + b 2 - k 2 M C × M D = a 2 + b 2 - k 2

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