NIMCET 2009 — Mathematics PYQ

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Question

NIMCET 2009 — Mathematics PYQ

NIMCET | Mathematics | 2009

The straight lines $\frac{x}{a} - \frac{y}{b} = k$ and $\frac{x}{a} + \frac{y}{b} = \frac{1}{k}, k \neq 0$ meet on:

Choose the correct answer:

Explanation

1. Write the given equations:

Equation 1: $\frac{x}{a} - \frac{y}{b} = k$
Equation 2: $\frac{x}{a} + \frac{y}{b} = \frac{1}{k}$

2. Eliminate the parameter $k$ :

The easiest way to eliminate $k$ is to multiply the two equations together.

\left( \frac{x}{a} - \frac{y}{b} \right) \left( \frac{x}{a} + \frac{y}{b} \right) = k \cdot \frac{1}{k}

3. Simplify the expression:

Applying the algebraic identity $(A - B)(A + B) = A^2 - B^2$ :

\left( \frac{x}{a} \right)^2 - \left( \frac{y}{b} \right)^2 = 1

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

4. Identify the Locus:

The resulting equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is the standard equation of a hyperbola.

Final Answer:

The lines meet on a hyperbola. The correct option is (c).