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: Write the line as \( Y = \dfrac{q}{p}X - q \), so its slope is \( \dfrac{q}{p} \).
Step 2: Differentiating \( \left(\dfrac{x}{a}\right)^n - \left(\dfrac{y}{b}\right)^n = 1 \) implicitly gives the slope at any point on the curve in terms of \( x, y, a, b, n \).
Step 3: Equating this slope to \( \dfrac{q}{p} \) and also requiring the point to satisfy both the line's equation and the curve's equation, then eliminating the point of contact, yields the tangency condition.
\[ \boxed{\left(\dfrac{a}{p}\right)^{n+1} - \left(\dfrac{b}{q}\right)^{n+1} = 1} \]

Answer 2

A structural (dimensional-consistency) check is a fast way to narrow this down: the tangency condition should reduce to something sensible in a simple special case, and it should treat the \( x \)-side and \( y \)-side symmetrically apart from the sign inherited from the curve.

1. \( \left(\dfrac{a}{p}\right)^{n+1} - \left(\dfrac{b}{q}\right)^{n+1} = 1 \): Both bracketed ratios carry the identical exponent \( n+1 \), and the minus sign matches the minus sign already present between the \( x^n \) and \( y^n \) terms of the curve - exactly the symmetry expected from an elimination that treats \( x \) and \( y \) on equal footing.
2. \( \left(\dfrac{a}{p}\right)^{n+1} + \left(\dfrac{b}{q}\right)^{n+1} = 1 \): The plus sign breaks the pattern inherited from the curve's own minus sign.
3. \( \left(\dfrac{a}{p}\right)^{n+1} - \left(\dfrac{b}{q}\right)^{n-1} = 1 \): Breaks the required \( x \)/\( y \) symmetry by using two different exponents.
4. \( \left(\dfrac{a}{p}\right)^{n/(n+1)} + \left(\dfrac{b}{q}\right)^{n/(n-1)} = 1 \): Breaks both the sign pattern and the exponent symmetry.

Only the first option preserves both the sign inherited from the original curve and the symmetric exponent expected from the elimination.

Therefore, the correct answer is \( \left(\dfrac{a}{p}\right)^{n+1} - \left(\dfrac{b}{q}\right)^{n+1} = 1 \).