Find the relation between $p$ and $q$ by simplifying the expression $(6 p + 5 q)^{2} + (6 p - 5 q)^{2}$ .
Also, find the value of expression when $p^{2} = \frac{4}{9}$ and $q^{2} = \frac{2}{5}$ .

2 (36 p^{2} + 25 q^{2})

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52

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36 p^{2} + 25 q^{2}

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26

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Solution

The correct option is B $52$
Given the expression
$(6 p + 5 q)^{2} + (6 p - 5 q)^{2}$
Let $(6 p + 5 q) = x$ and $(6 p - 5 q) = y$
$∴$ the expression becomes $x^{2} + y^{2}$
$\Rightarrow x^{2} + y^{2} = (x - y)^{2} + 2 x y$
$[∵ (a - b)^{2} = a^{2} + b^{2} - 2 a b]$
Now, re-substituting value of $x$ and $y$ in the above expression we get,
$\Rightarrow x^{2} + y^{2} = [(6 p + 5 q) - (6 p - 5 q)]^{2} + 2 (6 p + 5 q) (6 p - 5 q)$
Applying the formula $(a - b) (a + b) = (a^{2} - b^{2})$ we get,
$\Rightarrow x^{2} + y^{2} = [6 p + 5 q - 6 p + 5 q]^{2} + 2 [(6 p)^{2} - (5 q)^{2}]$
$\Rightarrow x^{2} + y^{2} = (10 q)^{2} + 2 [36 p^{2} - 25 q^{2}]$
$\Rightarrow x^{2} + y^{2} = 100 q^{2} + 72 p^{2} - 50 q^{2}$
$\Rightarrow x^{2} + y^{2} = 50 q^{2} + 72 p^{2} = 2 (25 q^{2} + 36 p^{2})$
So, this is the required relarion between $p$ and $q$ .
Now,Putting value of $p^{2} = \frac{4}{9}$ and $q^{2} = \frac{2}{5}$ we get
$(6 p + 5 q)^{2} + (6 p - 5 q)^{2} = 50 q^{2} + 72 p^{2} = 50 \times \frac{2}{5} + 72 \times \frac{4}{9}$
$\Rightarrow 50 q^{2} + 72 p^{2} = 20 + 32 = 52$

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