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

Answer

To solve the problem, we need to find the value of \( (4k)^6 \) given that the curves \( x = y^4 \) and \( xy = k \) intersect at right angles.

### Step-by-Step Solution:

1. **Identify the curves:**
The first curve is given by:
\[ x = y^4 \]
The second curve is:
\[ xy = k \]

2. **Substitute \( x \) from the first curve into the second curve:**
Substitute \( x = y^4 \) into \( xy = k \):
\[ y^4 \cdot y = k \implies y^5 = k \]
Thus, we have:
\[ k = y^5 \quad \text{(Equation 1)} \]

3. **Find the slope of the tangent for the first curve:**
Differentiate \( x = y^4 \) with respect to \( y \):
\[ \frac{dx}{dy} = 4y^3 \]
To find \( \frac{dy}{dx} \), we take the reciprocal:
\[ \frac{dy}{dx} = \frac{1}{4y^3} \quad \text{(Slope of the first curve, } M_1\text{)} \]

4. **Find the slope of the tangent for the second curve:**
Differentiate \( xy = k \) using the product rule:
\[ x \frac{dy}{dx} + y = 0 \implies \frac{dy}{dx} = -\frac{y}{x} \]
Substitute \( x = y^4 \) into this equation:
\[ \frac{dy}{dx} = -\frac{y}{y^4} = -\frac{1}{y^3} \quad \text{(Slope of the second curve, } M_2\text{)} \]

5. **Set the condition for perpendicularity:**
For the curves to intersect at right angles, the product of their slopes must be \(-1\):
\[ M_1 \cdot M_2 = -1 \]
Substitute the slopes:
\[ \left(\frac{1}{4y^3}\right) \left(-\frac{1}{y^3}\right) = -1 \]
Simplifying gives:
\[ -\frac{1}{4y^6} = -1 \implies \frac{1}{4y^6} = 1 \implies 4y^6 = 1 \implies y^6 = \frac{1}{4} \]

6. **Find \( y \):**
Taking the sixth root:
\[ y = \left(\frac{1}{4}\right)^{\frac{1}{6}} = \frac{1}{2^{\frac{2}{3}}} \]

7. **Substitute \( y \) back to find \( k \):**
Using \( k = y^5 \):
\[ k = \left(\frac{1}{2^{\frac{2}{3}}}\right)^5 = \frac{1}{2^{\frac{10}{3}}} \]

8. **Calculate \( (4k)^6 \):**
First, find \( 4k \):
\[ 4k = 4 \cdot \frac{1}{2^{\frac{10}{3}}} = \frac{4}{2^{\frac{10}{3}}} = \frac{2^2}{2^{\frac{10}{3}}} = 2^{2 - \frac{10}{3}} = 2^{\frac{6}{3} - \frac{10}{3}} = 2^{-\frac{4}{3}} \]
Now calculate \( (4k)^6 \):
\[ (4k)^6 = \left(2^{-\frac{4}{3}}\right)^6 = 2^{-8} = \frac{1}{256} \]

### Final Answer:
Thus, the value of \( (4k)^6 \) is:
\[ \frac{1}{256} \]