JEE MAIN - Mathematics (2024 - 5th April Morning Shift - No. 15)
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
12
13
11
10
Explanation
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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
$$\begin{aligned} & x^2+9 y^2=k^2=6^2 \\ & \Rightarrow \frac{x^2}{6^2}+\frac{y^2}{2^2}=1 \\ & \text { L.R }=\frac{2 b^2}{a}=\frac{2 \times 4}{6}=\frac{4}{3} \\ & m=4 \\ & n=3 \\ & 2 m+n=8+3=11 \end{aligned}$$
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