Question

The equation of the tangent at \( (2, -3) \) on the hyperbola \( x^2 - \frac{y^2}{3} = 1 \) is:

• A. \( x + 2y = 0 \)
• B. \( 2x + y = 0 \)
• C. \( x + 2y = 1 \)
• D. \( 2x + y = 1 \)

Hint:

To find tangent at point \( (x_1, y_1) \) on hyperbola: \[ \text{Use: } \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 \text{ for } \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

Answer 1

The Correct Option is D

Step 1: Identify the general form of hyperbola.
Given hyperbola:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] Compare with: \[ x^2 - \frac{y^2}{3} = 1 \Rightarrow a^2 = 1, b^2 = 3 \] Step 2: Use the equation of tangent at \( (x_1, y_1) \).
Standard tangent to hyperbola:
\[ \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 \] Substitute values \( (x_1, y_1) = (2, -3) \), \( a^2 = 1, b^2 = 3 \): \[ \frac{x \cdot 2}{1} - \frac{y \cdot (-3)}{3} = 1 \Rightarrow 2x + y = 1 \]