The area (in square units) of the region bounded by the parabola y 2 = 4 ( x - 2 ) and the line y = 2 x - 8 , is [2024]

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Solution

Q.
The area (in square units) of the region bounded by the parabola $y^{2} = 4 (x - 2)$ and the line $y = 2 x - 8,$ is [2024]

1 9

2 7

3 8

4 6

Ans.
(1)

Parabola : $y^{2} = 4 (x - 2)$ ...(i)

and line : $y = 2 x - 8$ ...(ii)

$\Rightarrow 4 {(x - 4)}^{2} = 4 (x - 2)$

$\Rightarrow x^{2} - 9 x + 18 = 0$

$\Rightarrow x = 3, 6$

From (ii), we get

$\Rightarrow y = - 2, 4$

Points of intersection of (i) and (ii) are $(3, - 2)$ and $(6, 4)$

$∴ Required Area = \int_{- 2}^{4} [(\frac{y + 8}{2}) - (\frac{y^{2}}{4} + 2)] d y$

$= {[\frac{y^{2}}{4} - \frac{y^{3}}{12} + 2 y]}_{- 2}^{4}$

$= 9 sq. units$

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