Let the line [2 x+3 y-k=0, k0], intersect the x-axis and y-axis at the points [A] and [B]…

Question

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 $A B$ 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

Accepted Answer

Equation of circle with $A B$ as diameter

$$\begin{aligned}
& \left(x-\frac{k}{2}\right) x+y\left(y-\frac{k}{3}\right)=0 \
& \Rightarrow x^2+y^2-\frac{k x}{2}-\frac{k y}{3}=0
\end{aligned}$$

Comparing, $k=6$

Latus rectum of ellipse

$$\begin{aligned}
& x^2+9 y^2=k^2=6^2 \
& \Rightarrow \frac{x^2}{6^2}+\frac{y^2}{2^2}=1 \
& \text { L.R }=\frac{2 b^2}{a}=\frac{2 \times 4}{6}=\frac{4}{3} \
& m=4 \
& n=3 \
& 2 m+n=8+3=11
\end{aligned}$$

Solutions · 1