Question

If the curves $x=y^4$ and $xy=k$ cut at right angles, then $(4k)^6$ is equal to ________ .

Hint:

For orthogonal curve problems, the process is: 1. Find the slopes ($m_1, m_2$) of both curves. 2. Set $m_1 \cdot m_2 = -1$ at the intersection point $(x_0, y_0)$. 3. Use the original curve equations to substitute and eliminate $x_0, y_0$ to find the required constant.

Answer 1

Correct Answer: 4

Curves: \[ x=y^4,\quad xy=k \] Slopes: \[ m_1=\frac{1}{4y^3},\quad m_2=-\frac{y}{x} \] Orthogonality: \[ m_1m_2=-1 \Rightarrow 4xy^2=1 \] Since \(x=y^4\): \[ 4y^6=1 \Rightarrow y^6=\frac14 \] \[ k=xy=y^5 \] \[ (4k)^6=4^6(y^6)^5=\frac{4^6}{4^5}=4 \] \[ \boxed{4} \]