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

Question

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

(1) 37

(2) 46

(3) 58

(4) 72

Answer 1

Correct option is (3) 58  

Equation of chord \(T=S_{1}\)

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

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

\(\Rightarrow 40 x+18 y=109\)

\(\Rightarrow \alpha=40, \beta=18\)

\(\Rightarrow \alpha+\beta=58\)