Let the line $2 x+3 y-\mathrm{k}=0, \mathrm{k}>0$, intersect the $x$-axis and $y$-axis at the points…
Let the line $2 x+3 y-\mathrm{k}=0, \mathrm{k}>0$, intersect the $x$-axis and $y$-axis at the points $\mathrm{A}$ and $\mathrm{B}$, respectively. If the equation of the circle having the line segment $\mathrm{AB}$ as a diameter is $x^2+y^2-3 x-2 y=0$ and the length of the latus rectum of the ellipse $x^2+9 y^2=k^2$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $2 \mathrm{~m}+\mathrm{n}$ is equal to
11
10
12
13
Solution
Centre of the circle $=\left(\frac{3}{2}, 1\right)$
Equation of diameter $=2 \mathrm{x}+3 \mathrm{y}-\mathrm{k}=0$
$\begin{aligned}
& 2\left(\frac{3}{2}\right)+3(1)-\mathrm{k}=0 \\
& \Rightarrow \mathrm{k}=6
\end{aligned}$
Now, Equation of ellipse becomes
$\begin{aligned}
& x^2+9 y^2=36 \\
& \frac{x^2}{6^2}+\frac{y^2}{2^2}=1
\end{aligned}$
$\begin{aligned} & \text { length of } L R=\frac{2 b^2}{a}=\frac{2.2^2}{6}=\frac{8}{6}=\frac{4}{3}=\frac{m}{n} \\ & \therefore 2 m+n=2(4)+3=11\end{aligned}$