Question

The acute angle between the curve x = 2y² and y = 2x² at `(1/2, 1/2)` is ______.

Options

• `tan^-1 (4/5)`

• `tan^-1 (3/4)`

• `tan^-1(1)`

• `tan^-1 (4/3)`

Answer 1

The acute angle between the curve x = 2y² and y = 2x² at `(1/2, 1/2)` is `tan^-1 (3/4)`.

Explanation:

x = 2y² 

∴ 1 = `4y ("d"y)/("d"x)`

∴ `(("d"y)/("d"x))_(((1/2, 1/2))) = 1/2` = m₁  .....(say)

and y = 2x² 

∴ `("d"y)/("d"x)` = 4x

∴ `(("d"y)/("d"x))_(((1/2, 1/2)))` = 2 = m₂  .....(say)

∴ Angle of intersection is tan θ = `|("m"_1 - "m"_2)/(1 + "m"_1"m"_2)|`

= `|(1/2 - 2)/(1 + (1/2)2)|`

= `3/4`

∴ θ = `tan^-1 (3/4)`