Question

Let T be the tangent to the ellipse E: x² + 4y² = 5 at the point P(1, 1). If the area of the region bounded by the tangent T, ellipse E, lines x = 1 and x = `sqrt(5)` is `sqrt(5)`α + β + γ `cos^-1(1/sqrt(5))`, then |α + β + γ| is equal to ______.

Options

• 1.24

• 1.25

• 1.26

• 1.27

Answer 1

Let T be tangent to the ellipse E: x² + 4y² = 5 at the point P(1, 1). If the area of the region bounded by the tangent T, ellipse E, lines x = 1 and x = `sqrt(5)` is `sqrt(5)`α + β + γ `cos^-1(1/sqrt(5))`, then |α + β + γ| is equal to 1.25.

Explanation:

Equation of ellipse x² + 4y² = 5

or, `x^2/5 + (4y^2)/5` = 1

Equation of tangent at P(1, 1) is x + 4y = 5

Now, area bounded by the required region = `int_1^sqrt(5)(((5 - x)/4) - sqrt((5 - x^2)/4))dx`

= `(5/4x - x^2/8)|""_1^sqrt(5)- 1/2[x/2 sqrt(5 - x^2) + 5/2sin^-1(x/sqrt(5))]|_1^sqrt(5)`

= `((5sqrt(5))/4 - 5/8) - (5/4 - 1/8) - 1/2(0 + (5π)/4) + 1/2(1 + 5/2sin^-1(1/sqrt(5)))`

= `5/4(sqrt(5) - 1) - 1/2 - (5π)/8 + 1/2 + 5/4sin^-1(1/sqrt(5))`

= `(5sqrt(5))/4 - 5/4 - 5/4cos^-1(1/sqrt(5))`

Comparing the given condition

⇒ α = `5/4`, β = `(-5)/4` and γ = `(-5)/4`

⇒ |α +  β + γ| = `5/4` = 1.25