A radius-1 circle C touches the x -axis at A , with centre Q in the first quadrant. A tangent from O = ( 0 , 0 ) touches it at T . A point P on this tangent makes △ O A P right-angled at A with perime

Question

A radius-1 circle C touches the x -axis at A , with centre Q in the first quadrant. A tangent from O = ( 0 , 0 ) touches it at T . A point P on this tangent makes △ O A P right-angled at A with perimeter 8. Find QP .

TestHub · Accepted Answer

Let O A = u , A P = v , OP = w . Tangency of line OP to the unit circle centred at ( u , 1 ) gives u ( v − 1 ) / w = 1 ; also w 2 = u 2 + v 2 and u + v + w = 8 . Solving gives u = 2 , v = 8/3 , w = 10/3 . Since Q = ( 2 , 1 ) , QP = 5/3 .

Mathematics - Circle Question with Solution | TestHub

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