Animated Solution for Mathematics - Differentiation: If the curves $x=y^4$ and $xy=k$ cut at right angles, then $(4k{)}^6$ is equal to

Welcome, future engineer. Today, we are not just solving a problem; we are exploring the elegant dance between two curves. When we look at the equations $x=y^4$ and $xy=k$, we aren't just looking at algebraic expressions.

We are looking at a sideways parabola and a rectangular hyperbola, two distinct geometric entities meeting at a point $P(x,y)$. The problem asks us to consider the case where these curves cut at right angles—a condition known as orthogonality.

This is a beautiful concept. It means that at the very moment they touch, their paths are perpendicular. Let us unravel this step by step.

To understand the intersection, we must understand the direction of each curve at point $P$. We need the slopes of the tangents.

For our first curve, $x=y^4$, we differentiate with respect to $y$:

Since the slope $m_1$ is defined as $\frac{dy}{dx}$, we take the reciprocal:

Now, consider the second curve, $xy=k$. To find its slope $m_2$, we use implicit differentiation with respect to $x$:

We now have the "DNA" of both curves at the point of intersection.

If two curves are orthogonal, their tangents are perpendicular. In the coordinate plane, this implies that the product of their slopes must be $-1$.

Setting $m_1\cdot m_2=-1$, we substitute our expressions:

The negative signs cancel out, and the $y$ terms simplify to $y^2$ in the denominator. We are left with the following constraint:

We need to evaluate the expression $(4k{)}^6$. Since $k=xy$, our target is $(4xy{)}^6$, which expands to $4^6{x}^6{y}^6$.

We know from our first curve that $x=y^4$. Substituting this into our constraint $4x{y}^2=1$:

Now, we express $x^6$ in terms of $y$. Since $x=y^4$, then $x^6=(y^4{)}^6=y^{24}$.

Substituting this into our target expression $4^6{x}^6{y}^6$:

We rewrite $y^{30}$ as $(y^6{)}^5$. Substituting $y^6=\frac{1}{4}$ into the expression:

The final result is 4.