$(a)\;p-\sqrt{p^2+q^2} : p+\sqrt{p^2+q^2}\qquad(b)\;p+\sqrt{p^2+q^2} : p-\sqrt{p^2+q^2}\qquad(c)\;p : q\qquad(d)\;p+\sqrt{p^2-q^2} : p-\sqrt{p^2-q^2}$

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Answer : (d) $\;p+\sqrt{p^2-q^2} : p-\sqrt{p^2-q^2}$

Explanation : AM : GM = p : q

$\large\frac{x+y}{2} : \sqrt{xy} = p : q$

$\large\frac{\large\frac{x+y}{2}}{\sqrt{xy}}=\large\frac{p}{q}$

$\large\frac{x+y}{2p}=\large\frac{\sqrt{xy}}{q}=k (say)$

$x+y=2kp----(1)$

$(x+y)^2=4k^2p^2$

$(x-y)^2=(x+y)^2-4xy$

$=(2kp)^2-4(kq)^2$

$x-y=2k\;\sqrt{p^2-q^2}-----(2)$

$2x=2kp+2k\;\sqrt{p^2-q^2}$

$2x=2kp-2k\;\sqrt{p^2-q^2}$

$x : y= p+ \sqrt{p^2-q^2}: p-\sqrt{p^2-q^2}\;.$

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