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Question

XP and XQ are tangents from X to the circle with centre O.R is a point on the circle. Prove that, XA+AR=XB+BR.

Solution
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From the figure, there is an external point X from where two tangents, XP and XQ, are drawn to the circle. XP=XQ (The lengths of the tangents drawn from an external point to the circle are equal.)Similarly,AP=ARBQ=BRXP=XA+AP --------- (1)XQ=XB+BQ --------- (2)By substituting AP=AR in equation (1) and BQ=BR in equation (2), we getXP=XA+ARXQ=XB+BRSince the tangents XP and XQ are equal, we getXA+AR=XB+BR.

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