Applications of Derivatives
Tangent to Curves
Grade 12
Question:
<p>The equation of tangent to the curve \(\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2\) at \((a, b)\) is</p>
<p>(a) \(\frac{x}{a} + \frac{y}{b} = 2\)</p>
<p>(b) \(\frac{x}{a} + \frac{y}{b} = 1\)</p>
<p>(c) \(\frac{x}{a} - \frac{y}{b} = 2\)</p>
<p>(d) \(ax + by = 2\)</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation to find the slope at the given point, then write the tangent line equation.
<p>Using implicit differentiation on $\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$:</p><p>$\frac{n}{a}\left(\frac{x}{a}\right)^{n-1} + \frac{n}{b}\left(\frac{y}{b}\right)^{n-1}\frac{dy}{dx} = 0$</p><p>At point $(a,b)$: slope $= -\frac{b}{a}$</p><p>Tangent equation: $y - b = -\frac{b}{a}(x - a)$</p><p>Simplifying: $\frac{x}{a} + \frac{y}{b} = 2$</p>
Correct Answer: A