Question

If the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) having \((1,1)\) as its middle point is \( x + \alpha y = \beta \), then:

• A. \( \alpha + \beta = 1 \)
• B. \( \alpha + 1 = \beta \)
• C. \( \alpha - 1 = \beta \)
• D. \( 2\alpha - 1 = 3\beta \)

Hint:

In problems involving conics, always ensure the geometric properties (like center or foci) align with the algebraic properties you derive from equations.

Answer 1

The Correct Option is B

Step 1: Consider the equation of the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \). 
Step 2: Knowing the midpoint of the chord \((1,1)\) is on the ellipse, the line \(x + \alpha y = \beta\) must pass through this point. 
Step 3: Plugging \((1,1)\) into the line equation gives: \[ 1 + \alpha \times 1 = \beta \quad \Rightarrow \quad \beta = 1 + \alpha. \]