Question

If \(\alpha x + \beta y = 109\) is the equation of the chord of the ellipse \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \] whose midpoint is \(\left(\frac{5}{2}, \frac{1}{2}\right)\), then \(\alpha + \beta\) is equal to:

• A. 37
• B. 46
• C. 58
• D. 72

Hint:

For ellipse chord problems, using the midpoint formula simplifies the process of finding the correct equation of the chord.

Answer 1

The Correct Option is C

Step 1: Equation of the chord. The equation of the chord with midpoint \((h, k)\) is: \[ T = S_1 \] Where \( T = \frac{5x}{18} + \frac{y}{8} \quad \text{and} \quad S_1 = \frac{100 + 9}{144} = 109 \) Expanding the equation: \[ 40x + 18y = 109 \] Comparing with \(\alpha x + \beta y = 109\), we get: \[\alpha = 40, \quad \beta = 18\]

Step 2: Final Calculation. \[ \alpha + \beta = 40 + 18 = 58 \]