If the curves x = y⁴ and xy = k cut at right angles, then (4k)⁶ is equal to __________.

Solution

Maths sir · 4 steps

Get both slopes at the meeting point, set their product to -1, and 4k falls out as 1/y.

1. Two slopes. For x = y⁴, dx/dy = 4y³, so m₁ = 1/4y³. For xy = k, differentiating gives y + xdy/dx = 0, so m₂ = -y/x.

   m₁ m₂ = 1/4y³×(-y/x) = -1 ⇒ 4xy² = 1

2. Bring in k. At the meeting point xy = k, so 4xy² = 4k · y = 1. That gives y = 1/4k.
3. Use the first curve. Put x = y⁴ into 4xy² = 1 to get 4y⁶ = 1, so y⁶ = 1/4.
4. So. (4k)⁶ = 1/y⁶ = 4. The answer is 4.

The idea to keep. Curves cut at right angles means the product of their slopes at the meeting point is -1. Then use both curve equations at that point to remove x and y.

Answer from the NTA final answer key. Solution by PrepWiser.