If the area of the larger portion bounded between the curves and is , then b + c is equal to __________. [2025]

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Solution

Q.
If the area of the larger portion bounded between the curves $x^{2} + y^{2} = 25$ and $y = | x - 1 |$ is $\frac{1}{4} (b π + c), b, c \in N$ , then b + c is equal to __________. [2025]

Ans.
(77)

Given, $x^{2} + y^{2} = 25$

$\Rightarrow x^{2} + {(x - 1)}^{2} = 25$ [ $∵$ y = |x –1|]

$\Rightarrow x = 4, - 3$

$∴$ Required area $= 25 π - (\int_{- 3}^{4} (\sqrt{25 - x^{2}} - | x - 1 |) d x)$

$= 25 π - {[\frac{1}{2} x \sqrt{25 - x^{2}} + \frac{25}{2} \sin^{- 1} \frac{x}{5}]}_{- 3}^{4} + \int_{- 3}^{1} (1 - x) d x + \int_{1}^{4} (x - 1) d x$

$= 25 π + \frac{25}{2} - (2 \times 3 + \frac{25}{2} \sin^{- 1} (\frac{4}{5}) + \frac{3}{2} \times 4 + \frac{25}{2} \sin^{- 1} (\frac{3}{5}))$

$= 25 π + \frac{25}{2} - 12 - \frac{25 π}{4}$

$= \frac{75 π}{4} + \frac{1}{2} = \frac{1}{4} (75 π + 2)$

$∴ \frac{1}{4} (b π + c) = \frac{1}{4} (75 π + 2)$

$\Rightarrow b = 75, c = 2$

$\Rightarrow b + c = 77$ .

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