Question

The sum of the intercepts made by a tangent drawn to the curve \((x/a)^n + (y/b)^n = 2\) at \((a,b)\) on the coordinate axes is

• A. \(a+b\)
• B. \(a^2 + b^2\)
• C. \(2(a-b)\)
• D. \(2(a+b)\)

Hint:

For a curve \( (x/a)^n + (y/b)^n = \text{const} \), tangent at point \((a,b)\) in linear form gives intercepts; sum = sum of intercepts on axes.

Answer 1

The Correct Option is D

Step 1: Equation of tangent.
For curve \((x/a)^n + (y/b)^n = 2\), tangent at \((a,b)\) using derivative form: \[ \frac{x}{a} + \frac{y}{b} = 2 \quad \text{(since at point (a,b))} \]

Step 2: Find x-intercept.
Set \(y=0\): \(x/a = 2 \implies x = 2a\)

Step 3: Find y-intercept.
Set \(x=0\): \(y/b = 2 \implies y = 2b\)

Step 4: Sum of intercepts.
Sum = \(2a + 2b = 2(a+b)\)

Step 5: Verification.
Check for consistency with tangent form and intercepts formula

Step 6: Final conclusion.
Hence, sum of intercepts is \[ \boxed{2(a+b)} \]