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# The sum of the squares of two numbers is 97 and the squares of their difference is 25. The product of the two numbers is

 A) 45 B) 36 C) 54 D) 63

 B) 36

Let the two numbers are x and y.

According to the question,

Sum of squares fo two numbers = 97

ie., $$\Large x^{2} + y^{2} = 97$$ ... (i)

and the square of their difference = 25

ie., $$\Large \left(x - y\right)^{2} = 25$$ .. (ii)

=> $$\Large x - y = 5$$ .... (iii)

From eqn (ii)

$$\Large \left(x^{2} + y^{2}\right)-2xy = 25$$

=> 97 - 2xy = 25 (From eqn 1)

=> 2xy = 72

=< xy = 36 ... (iv)

Now, we have

$$\Large \left(x + y\right)^{2}= \left(x^{2} + y^{2} \right) + 2xy$$

= 97 + 72 = 169

$$\Large x + y =13$$ ... (V)

Now, from eqns (III) and (V) , we get

2x = 18

x = 9 and y = 4

The product of both the numbers  = xy = 9 X 4 = 36

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