Question

If the straight line \( \frac{x}{p} - \frac{y}{q} = 1 \) touches the curve \( \left( \frac{x}{a} \right)^n - \left( \frac{y}{b} \right)^n = 1 \), then

• A. \( \left( \frac{a}{p} \right)^{n+1} - \left( \frac{b}{q} \right)^{n+1} = 1 \)
• B. \( \left( \frac{a}{p} \right)^{n+1} + \left( \frac{b}{q} \right)^{n+1} = 1 \)
• C. \( \left( \frac{a}{p} \right)^{n+1} - \left( \frac{b}{q} \right)^{n-1} = 1 \)
• D. \( \left( \frac{a}{p} \right)^{n/n+1} + \left( \frac{b}{q} \right)^{n/n-1} = 1 \)

Hint:

For the tangency condition of curves and lines, the relationship between the coefficients of the curve and the line must satisfy the tangency condition.

Answer 1

The Correct Option is A

Step 1: Understanding the geometry of the tangent.
For a straight line to touch the curve, it must satisfy the condition that the distance between the line and the curve is zero. This gives us the relationship between the coordinates of the points on the curve and the line. We can use the condition of tangency to derive the equation. Step 2: Deriving the condition.
By substituting the values into the equation and simplifying, we find that the equation for the condition of tangency is: \[ \left( \frac{a}{p} \right)^{n+1} - \left( \frac{b}{q} \right)^{n+1} = 1 \] Step 3: Conclusion.
The correct equation is (1) \( \left( \frac{a}{p} \right)^{n+1 - \left( \frac{b}{q} \right)^{n+1} = 1 \)}.