Question

For what value of $ n $, the curve \[ \left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2 \text{ touches the straight line } \frac{x}{a} + \frac{y}{b} = 2 \text{ at the point } (a, b)? \]

• A. \( n = 3 \)
• B. Any value of \( n \)
• C. \( n = 2 \)
• D. \( n = 4 \)

Hint:

For curves involving powers like \( n \), the condition of tangency is satisfied when \( n = 2 \), which represents an ellipse or a degenerate form of a conic.

Answer 1

The Correct Option is C

Step 1:
The equation of the curve is given by:
\[ \left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2 \] We are asked to find the value of \( n \) such that this curve touches the straight line \( \frac{x}{a} + \frac{y}{b} = 2 \) at the point \( (a, b) \). 
Step 2:
Substituting the point \( (a, b) \) into the equation of the curve: \[ \left( \frac{a}{a} \right)^n + \left( \frac{b}{b} \right)^n = 2 \] This simplifies to: \[ 1^n + 1^n = 2 \] which holds true for any \( n \). 
Step 3:
To ensure the curve touches the straight line at the point \( (a, b) \), the curve must be in a form that can represent a degenerate conic section (like an ellipse or hyperbola) that meets the straight line exactly at one point. This is achieved when \( n = 2 \), which corresponds to the equation of an ellipse, and the curve touches the line. 
Step 4:
Therefore, the correct value of \( n \) is \( n = 2 \).