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See the attachment.
The centres are at C1 and C2.
the common chord is PQ. we need to prove PQ=2y.

radius of C1(C1P) = x
radius of C2(C2P) = y
distance between the centre (C1C2) = d
given that d² = x² - y²
⇒ d² + y² = x²

So C1C2P is a right angled triangle with C1P as hypotenuse. 
And C2P is the height. C2P is also the radius of the circle which is y. PQ is the diameter of the circle. So PQ = 2×C2P = 2y.
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