Question

The shortest distance between the line y = x and the curve y² = x – 2 is ______.

Options

• `11/4sqrt(2)`

• 2

• `7/4sqrt(2)`

• `7/8`

Answer 1

The shortest distance between the line y = x and the curve y² = x – 2 is `underlinebb(7/4sqrt(2))`.

Explanation:

y = x line and curve y² = x – 2 tangent parallel to y² = x – 2 and line y = x

`2y(dy)/(dx)` = 1, `(dy)/(dx)` = 1

`(dy)/(dx) = 1/(2y)`

`1/(2y)` = 1

y = `1/2`

On curve y² = x – 2

⇒ x = `9/4`

Tangent, `y - 1/2 = 1(x - 9/4)`

x – y = `7/4`

Distance between line and curve

x – y = `7/4`

`(7/4 - 0)/sqrt(1 + 1) = 7/(4sqrt(2))`