Question

In the figure ,two circle of radius xcm and ycm (x>y) intersect at two points P and Q respectively . if the distance 'd ' between the centre of two circles is given by d 2 =x 2 -y 2 prove that the length of the commond chord is 2ycm

Answer 1

Let the centers be    R and S respectively.
Join PQ which is perpendicular to  RS.  Let their intersection point be O.
In ΔRPO,  PO² + OR² = x²
In ΔSPO,  PO² + OS² = y²
Subtracting one from another:    OR² - OS² = x² - y²    ---(1)

Given:   RS² = (RO + OS)² = x² - y²
                         RO² + OS² + 2 RO * OS = x² - y²      ---(2)

(2) - (1):  2 OS (OS + RO) = 0
                   Hence  OS = 0,   as  OS ≠ - RO.

Then,  O and S are the same point.  So R, P , S form a right angle triangle at S.  Hence,   PS = half of common chord = radius of circle = y
 
Length of common chord = 2 y

Answer 2

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