If the area of the region { ( x , y ) : ? 1 - 2 x y 4 - x 2 , x 0 , y 0 } is , gcd = 1 , then the value of is: [2026]

Question

If the area of the region ${(x, y) : 1 - 2 x \leq y \leq 4 - x^{2}, x \geq 0, y \geq 0}$ is $\frac{α}{β}, α, β \in ℕ, \gcd (α, β) = 1$ , then the value of $(α + β)$ is: [2026]

Smart Achievers · Accepted Answer

(2)

$Required area = \frac{2}{3} \times 8 - \frac{1}{2} \times \frac{1}{2} \times 1$

$= \frac{16}{3} - \frac{1}{4} = \frac{61}{12} = \frac{α}{β}$

$\Rightarrow α + β = 73$

Solution

Q.
If the area of the region ${(x, y) : 1 - 2 x \leq y \leq 4 - x^{2}, x \geq 0, y \geq 0}$ is $\frac{α}{β}, α, β \in ℕ, \gcd (α, β) = 1$ , then the value of $(α + β)$ is: [2026]

1 91

2 73

3 85

4 67

Ans.
(2)

$Required area = \frac{2}{3} \times 8 - \frac{1}{2} \times \frac{1}{2} \times 1$

$= \frac{16}{3} - \frac{1}{4} = \frac{61}{12} = \frac{α}{β}$

$\Rightarrow α + β = 73$

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