If $\frac{7 m + 2 n}{7 m - 2 n} = \frac{5}{3}$ use properties of proportion to find $m : n$ and hence find $\frac{m^{2} - n^{2}}{m^{2} + n^{2}}$ .

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Solution

Given
$\frac{7 m + 2 n}{7 m - 2 n} = \frac{5}{3}$

By cross multiplying numerator and denominator,

$21 m + 6 n = 35 m - 10 n$

or, $14 m = 16 n$

$\frac{m}{n} = \frac{16}{14} = \frac{8}{7}$ ....... $(1)$

Now,
$\frac{m^{2} - n^{2}}{m^{2} + n^{2}}$

$= \frac{{(\frac{m}{n})}^{2} - 1}{{(\frac{m}{n})}^{2} + 1}$

$= \frac{8^{2} - 7^{2}}{8^{2} + 7^{2}}$ [Using $(1)$ ]

$= \frac{15}{113}$

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If 7m+2n7m−2n=53 use properties of proportion to find m:n and hence find m2−n2m2+n2.

Given7m+2n7m−2n=53By cross multiplying numerator and denominator,21m+6n=35m−10nor, 14m=16nmn=1614=87.......(1)Now,m2−n2m2+n2=(mn)2−1(mn)2+1=82−7282+72 [Using (1)]=15113

If $\frac{7 m + 2 n}{7 m - 2 n} = \frac{5}{3}$ use properties of proportion to find $m : n$ and hence find $\frac{m^{2} - n^{2}}{m^{2} + n^{2}}$ .