Question:
The equation of the tangent to the curve $\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$ at $(a, b)$ is
The equation of the tangent to the curve $\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$ at $(a, b)$ is
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At $(a,b)$, powers simplify nicely—use this to avoid heavy calculations.
Updated On: Apr 23, 2026
- $\frac{x}{a} + \frac{y}{b} = 2$
- $\frac{x}{a} + \frac{y}{b} = \frac{1}{2}$
- $\frac{x}{b} - \frac{y}{a} = 2$
- $ax + by = 2$
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The Correct Option is C
Solution and Explanation
Concept:
Implicit differentiation to find slope of tangent.
Step 1: Differentiate implicitly.
\[ \left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2 \] \[ n\left(\frac{x}{a}\right)^{n-1}\frac{1}{a} + n\left(\frac{y}{b}\right)^{n-1}\frac{1}{b}\frac{dy}{dx} = 0 \]
Step 2: Find slope at $(a,b)$.
\[ \left(\frac{a}{a}\right)^{n-1} = 1,\quad \left(\frac{b}{b}\right)^{n-1} = 1 \] \[ \frac{n}{a} + \frac{n}{b}\frac{dy}{dx} = 0 \] \[ \Rightarrow \frac{dy}{dx} = -\frac{b}{a} \]
Step 3: Equation of tangent.
\[ y - b = -\frac{b}{a}(x - a) \]
Step 4: Simplify.
\[ ay - ab = -bx + ab \] \[ bx + ay = 2ab \]
Step 5: Rearrange.
\[ \frac{x}{b} - \frac{y}{a} = 2 \] Conclusion:
Answer = Option (C)
Step 1: Differentiate implicitly.
\[ \left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2 \] \[ n\left(\frac{x}{a}\right)^{n-1}\frac{1}{a} + n\left(\frac{y}{b}\right)^{n-1}\frac{1}{b}\frac{dy}{dx} = 0 \]
Step 2: Find slope at $(a,b)$.
\[ \left(\frac{a}{a}\right)^{n-1} = 1,\quad \left(\frac{b}{b}\right)^{n-1} = 1 \] \[ \frac{n}{a} + \frac{n}{b}\frac{dy}{dx} = 0 \] \[ \Rightarrow \frac{dy}{dx} = -\frac{b}{a} \]
Step 3: Equation of tangent.
\[ y - b = -\frac{b}{a}(x - a) \]
Step 4: Simplify.
\[ ay - ab = -bx + ab \] \[ bx + ay = 2ab \]
Step 5: Rearrange.
\[ \frac{x}{b} - \frac{y}{a} = 2 \] Conclusion:
Answer = Option (C)
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