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

Question

Let the line $2x+3y-k=0,k>0$, intersect the $x$-axis and $y$-axis at the points $A$ and $B$, respectively. If the equation of the circle having the line segment $AB$ as a diameter is ${x}^{2}+{y}^{2}-3x-2y=0$ 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 $2m+n$ is equal to

Accepted Answer

Equation of circle with $AB$ as diameter

$\begin{gathered} (x-\frac{k}{2})x+y(y-\frac{k}{3})=0 \ \Rightarrow {x}^{2}+{y}^{2}-\frac{kx}{2}-\frac{ky}{3}=0 \end{gathered}$

Comparing, $k=6$

Latus rectum of ellipse

$\begin{gathered} {x}^{2}+9{y}^{2}={k}^{2}={6}^{2} \ \Rightarrow \frac{{x}^{2}}{{6}^{2}}+\frac{{y}^{2}}{{2}^{2}}=1 \ =\frac{2{b}^{2}}{a}=\frac{2	imes 4}{6}=\frac{4}{3} \ m=4 \ n=3 \ 2m+n=8+3=11 \end{gathered}$

Solutions · 1