Let the line x = - 1 divide the area of the region { ( x , y ) : 1 + x 2 y 3 - x } in the ratio m : n , gcd ( m , n ) = 1 . Then m + n is equal to [2026]

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Solution

Q.
Let the line $x = - 1$ divide the area of the region ${(x, y) : 1 + x^{2} \leq y \leq 3 - x}$ in the ratio $m : n$ , $\gcd (m, n) = 1$ . Then $m + n$ is equal to [2026]

1 26

2 25

3 28

4 27

Ans.
(4)

$\frac{m}{n} = \frac{\int_{- 1}^{1} [(3 - x) - (1 + x^{2})] d x}{\int_{- 2}^{1} [(3 - x) - (1 + x^{2})] d x} = \frac{20}{7}$

$∴$ $m + n = 20 + 7$

$= 27$

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