Question

The two curves x³ – 3xy² + 2 = 0 and 3x²y – y³ = 2 ______.

Options

• Touch each other

• Cut at right angle

• Cut at an angle `pi/3`

• Cut at an angle `pi/4`

Answer 1

The two curves x³ – 3xy² + 2 = 0 and 3x²y – y³ = 2 cut at right angle.

Explanation:

From first equation of the curve

We have 3x² – 3y² – 6xy `"dy"/"dx"` = 0

⇒ `"dy"/"dx" = (x^2 - y^2)/(2xy)` = (m₁) say and second equation of the curve gives

`6xy + 3x^2 "dy"/"dx" - 3y^2 "dy"/"dx"` = 0

⇒ `"dy"/"dx" = (-2y)/(x^2 - y^2)` = (m₂) say

Since m₁ . m₂ = –1.