If is the equation of the chord of the ellipse , whose mid point is , then is equal to : [2025]

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Solution

Q.
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 $(\frac{5}{2}, \frac{1}{2})$ , then $α + β$ is equal to : [2025]

1 46

2 58

3 37

4 72

Ans.
(2)

We have, equation of ellipse as, $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$

Equation of chord with mid-point $(\frac{5}{2}, \frac{1}{2})$ is

$\frac{x x_{1}}{a^{2}} + \frac{y y_{1}}{b^{2}} = \frac{x_{1}^{2}}{a^{2}} + \frac{y_{1}^{2}}{b^{2}}$

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

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

$\Rightarrow$ 40x + 18y = 109

On comparing with given equation $α x + β y = 109$ , we get

$α = 40 and β = 18$

$∴ α + β = 58$ .

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