Question

If $(x/a)^n + (y/b)^n = 2$, then $dy/dx$ at $(a,b)$ is

• A. $a/b$
• B. $-a/b$
• C. $b/a$
• D. $-b/a$

Hint:

For curves of form $(x/a)^n + (y/b)^n = C$, the slope at $(a,b)$ is always $-b/a$.

Answer 1

The Correct Option is D

Step 1: Concept Perform implicit differentiation on the given equation.

Step 2: Meaning Differentiate both sides with respect to $x$: $n(x/a)^{n-1}(1/a) + n(y/b)^{n-1}(1/b)(dy/dx) = 0$.

Step 3: Analysis Solve for $dy/dx$: $dy/dx = -[n(x/a)^{n-1}/a] / [n(y/b)^{n-1}/b] = -(b/a) \cdot (x/a)^{n-1} / (y/b)^{n-1}$.

Step 4: Conclusion At point $(a,b)$, substitute $x=a$ and $y=b$: $dy/dx = -(b/a) \cdot (1)^{n-1} / (1)^{n-1} = -b/a$. Final Answer: (D)