If x+ y=109 is the equation of the chord of the ellipse x^29+y^24=1, whose mid point is ((…

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 mid point is $(\frac{5}{2},\frac{1}{2})$. then $\alpha +\beta $ is equal to :

Accepted Answer

$$\begin{aligned}
&\text { Equation of chord } \mathrm{T}=\mathrm{S}_1\
&\begin{aligned}
& \frac{5}{2}\left(\frac{\mathrm{x}}{9}\right)+\frac{1}{2}\left(\frac{\mathrm{y}}{4}\right)=\frac{25}{36}+\frac{1}{16} \
& \Rightarrow \frac{5 \mathrm{x}}{18}+\frac{\mathrm{y}}{8}=\frac{100+9}{144}=\frac{109}{144} \
& \Rightarrow 40 \mathrm{x}+18 \mathrm{y}=109 \
& \Rightarrow \alpha=40, \beta=18 \
& \Rightarrow \alpha+\beta=58
\end{aligned}
\end{aligned}$$

Solutions · 1