Key Concepts and Formulas

• Orthogonal Intersection: Two curves intersect orthogonally if the product of their tangent slopes at the point of intersection is -1, i.e., $m_1 m_2 = -1$.
• Implicit Differentiation: Used to find the derivative of a function defined implicitly. If $y$ is a function of $x$, then $\frac{d}{dx}(y^n) = ny^{n-1}\frac{dy}{dx}$.
• Product Rule: The derivative of a product of two functions is given by $\frac{d}{dx}(uv) = u'v + uv'$.

Step-by-Step Solution

Step 1: Find the slopes of the tangents to each curve.

We need to find $\frac{dy}{dx}$ for both curves. This represents the slope of the tangent at any point $(x,y)$ on the curve.

Curve 1: $x = y^4$ Differentiate both sides with respect to $x$ using implicit differentiation: $\frac{d}{dx}(x) = \frac{d}{dx}(y^4)$ $1 = 4y^3 \frac{dy}{dx}$ Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{1}{4y^3}$ Let $m_1 = \frac{1}{4y^3}$. This is the slope of the tangent to the first curve.

Curve 2: $xy = k$ Differentiate both sides with respect to $x$ using the product rule and implicit differentiation: $\frac{d}{dx}(xy) = \frac{d}{dx}(k)$ $x\frac{dy}{dx} + y(1) = 0$ $x\frac{dy}{dx} = -y$ Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{y}{x}$ Since $xy = k$, we have $x = \frac{k}{y}$. Substitute this into the expression for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{y}{k/y} = -\frac{y^2}{k}$ Let $m_2 = -\frac{y^2}{k}$. This is the slope of the tangent to the second curve.

Step 2: Find the relationship between $y$ and $k$ at the point of intersection.

The curves intersect where their equations are simultaneously satisfied. $x = y^4$ $xy = k$ Substitute $x = y^4$ into the second equation: $(y^4)y = k$ $y^5 = k$

Step 3: Apply the orthogonality condition.

Since the curves intersect at right angles, the product of their slopes at the point of intersection is -1: $m_1 m_2 = -1$ Substitute the expressions for $m_1$ and $m_2$: $\left(\frac{1}{4y^3}\right) \left(-\frac{y^2}{k}\right) = -1$ $-\frac{y^2}{4ky^3} = -1$ $-\frac{1}{4ky} = -1$ $1 = 4ky$ $y = \frac{1}{4k}$

Step 4: Solve for $(4k)^6$.

From Step 2, we have $y^5 = k$. From Step 3, we have $y = \frac{1}{4k}$. Substitute the second equation into the first: $\left(\frac{1}{4k}\right)^5 = k$ $\frac{1}{(4k)^5} = k$ $1 = k(4k)^5$ $1 = k(4^5 k^5)$ $1 = 4^5 k^6$ Multiply both sides by 4: $4 = 4^6 k^6$ $4 = (4k)^6$

Thus, $(4k)^6 = 4$.

Common Mistakes & Tips

• Remember to use the chain rule correctly when performing implicit differentiation.
• Pay close attention to algebraic manipulations to avoid errors.
• Expressing both slopes in terms of a single variable (in this case, $y$) simplifies the subsequent calculations.

Summary

We found the slopes of the tangent lines for both curves using implicit differentiation. Then, using the condition for orthogonal intersection ($m_1 m_2 = -1$) and the relationship derived from the intersection of the two curves, we solved for the value of $(4k)^6$. The final answer is 4.

Final Answer The final answer is \boxed{4}.