A line passing through the point A(–2, 0), touches the parabola at the point B in the first quadrant. The area, of the region bounded by the line AB, parabola P and the x-axis, is : [2025]

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Solution

Q.
A line passing through the point A(–2, 0), touches the parabola $P : y^{2} = x - 2$ at the point B in the first quadrant. The area, of the region bounded by the line AB, parabola P and the x-axis, is : [2025]

1 $\frac{7}{3}$

2 2

3 3

4 $\frac{8}{3}$

Ans.
(4)

Given parabola $P : y^{2} = x - 2$ and line is passing through A(–2, 0)

Tangent is y = m(x + 2)

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

$\Rightarrow m^{2} x^{2} + (4 m^{2} - 1) x + (4 m^{2} + 2) = 0$

As D = 0

$\Rightarrow {(4 m^{2} - 1)}^{2} = 4 m^{2} (4 m^{2} + 2) \Rightarrow m = \frac{1}{4}$

So, tangent is $y = \frac{1}{4} (x + 2)$

Point of tangency is (6, 2)

$∴$ Area of region $= \int_{0}^{2} (y^{2} + 2) - (4 y - 2) d y$

$= \frac{8}{3} - 8 + 8 = \frac{8}{3}$ sq. units.

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