Question

The area of the region bounded by the parabola (y – 2)² = (x – 1), the tangent to it at the point whose ordinate is 3 and the x-axis is ______.

Options

• 9

• 10

• 4

• 6

Answer 1

The area of the region bounded by the parabola (y – 2)² = (x – 1), the tangent to it at the point whose ordinate is 3 and the x-axis is 9.

Explanation:

Since the ordinate is 3.

so, y = 3 `\implies` x = 2

Point of contact is (2, 3).

Given the parabola equation (y – 2)² = (x – 1).

Diff. w.r.t. x,

2 (y – 2)y' = 1

`\implies` y' = `1/(2(y - 2))`

`\implies` y'_(2,3) = `1/2`

`\implies` `(y - 3)/(x - 2) = 1/2`

`\implies` x – 2y + 4 = 0

The equation of tangent to the given parabola is x – 2y + 4 = 0.

Required area of the bounded region

= `int_0^3((y - 2)^2 + 1 - (2y - 4))dy`

= 9 sq. units