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# If AM and GM of two numbers are in ratio $p : q$, then the ratio of two numbers is

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

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A)
Answer : (d) $\;p+\sqrt{p^2-q^2} : p-\sqrt{p^2-q^2}$
Explanation : AM : GM = p : q
$\large\frac{x+y}{2} : \sqrt{xy} = p : q$
$\large\frac{\large\frac{x+y}{2}}{\sqrt{xy}}=\large\frac{p}{q}$
$\large\frac{x+y}{2p}=\large\frac{\sqrt{xy}}{q}=k (say)$
$x+y=2kp----(1)$
$(x+y)^2=4k^2p^2$
$(x-y)^2=(x+y)^2-4xy$
$=(2kp)^2-4(kq)^2$
$x-y=2k\;\sqrt{p^2-q^2}-----(2)$
$2x=2kp+2k\;\sqrt{p^2-q^2}$
$2x=2kp-2k\;\sqrt{p^2-q^2}$
$x : y= p+ \sqrt{p^2-q^2}: p-\sqrt{p^2-q^2}\;.$