Question

The equations of asymptotes for the hyperbola \(xy - 3x - 2y = 0\) are:

• A. \(x = 2, y = 3\)
• B. \(x = 2, y = 1\)
• C. \(x = 3, y = 1\)
• D. \(x = 1, y = 3\)

Hint:

Exam Tip:
For \(xy - hx - ky = 0\):

• Add \(hk\) to complete the square.
• \((x - k)(y - h) = hk\).
• Asymptotes are \(x = k\) and \(y = h\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to find the asymptotes of a hyperbola. The given equation is \(xy - 3x - 2y = 0\), which is a rectangular hyperbola.

Step 2: Key Formula or Approach:
For a hyperbola of the form \(xy - hx - ky = 0\), the asymptotes are \(x = k\) and \(y = h\).

Step 3: Detailed Explanation:
Rewrite the equation: \[ xy - 3x - 2y = 0 \] Add 6 to both sides: \[ xy - 3x - 2y + 6 = 6 \] \[ (x - 2)(y - 3) = 6 \] This is a rectangular hyperbola with center \((2, 3)\).
The asymptotes are given by \((x - 2)(y - 3) = 0\), i.e., \(x = 2\) and \(y = 3\).

Step 4: Final Answer:
Therefore, option (A) is correct.