If α x+β y=109 is the equation of the chord of the ellipse frac x 2 9 +frac y 2 4 =1 , whose mid point is (5/2, 1/2 ) , then α+β is equal to

Mathematics · Ellipse

JEE Main 2025 — 29 January, Evening Shift — Question 50

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

Option A:
37
Option B:
46
Option C:
58
Correct
Option D:
72

Answer: C

Step-by-step solution

Equation of chord $T=S_{1}$

$\frac{5}{2}\left(\frac{x}{9}\right)+\frac{1}{2}\left(\frac{y}{4}\right)=\frac{25}{36}+\frac{1}{16}$

$\Rightarrow \frac{5 x}{18}+\frac{y}{8}=\frac{100+9}{144}=\frac{109}{144}$

$\Rightarrow 40 x+18 y=109$

$\Rightarrow \alpha=40, \beta=18$

$\Rightarrow \alpha+\beta=58$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 29 January, Evening Shift
Subject: Mathematics
Chapter: Ellipse
Topic: Tangents & Normals to ellipse, chord of conatct