If $$a (a + b + c) = 45; b (a + b + c) = 75$$ and $$c ( a + b + c) = 105$$, then find the value of $$a^2 + b^2 + c^2$$.

ATQ

$$a\times (a + b + c)$$ = 45 (i)

$$b\times (a + b + c)$$ = 75  (ii)

$$c\times (a + b + c)$$= 105 (iii)

Adding (i), (ii) and (iii), we get:

   (a + b + c) (a + b + c)   =45 + 75 + 105

(a + b + c) (a + b + c) = 225

Taking square root on both sides of above eqn we get:

(a + b + c) = 15

Put the value of (a + b + c) in eqn.s (i), (ii) and (iii), we get:

$$a\times 15 = 45$$

$$b\times 15 = 75$$

$$c\times 15 = 105$$

so,

a = 3, b = 5, c = 7

So, we have

(a^2 + b^2 + c^2) = 3^2 + 5^2 + 7^2

= 9 + 25 + 49

=83.