Question

If curves y² = 4x and xy = c cut at right angles, then the value of c is ______.

Options

• `4sqrt(2)`

• 8

• `2sqrt(2)`

• `-4sqrt(2)`

Answer 1

If curves y² = 4x and xy = c cut at right angles, then the value of c is `underlinebb(4sqrt(2))`.

Explanation:

Given curves, y² = 4x and xy = c cuts orthogonally.

Let they intersect at (x₁, y₁)·

Now, y² = 4x

∴ `2y dy/dx = 4`

`\implies dy/dx = 2/y`

`\implies (dy)/(dx)|_((x_1","y_1)) = 2/y_1`  ...(i)

And xy = c

∴ `x dy/dx + y = 0`

∴ `dy/dx = (-y)/x`

`\implies (dy)/(dx)|_((x_1"," y_1)) = -y_1/x_1`  ...(ii)

From equations (i) and (ii)

`2/y_1 xx ((-y_1)/x_1) = -1`  ...[∵ m₁m₂ = –1]

`\implies` x₁ = 2

Put x₁ = 2 in y₁² = 4x₁, we get

y₁² = 4(2) = 8

`\implies y_1 = 2sqrt(2)`

 Now, put value of x₁ and y₁ in x₁y₁ = c, we get

`c = 2(2sqrt(2)) = 4sqrt(2)`