Ellipse
Latus Rectum from Circle and Line
nta_pyq_2024_apr
Grade 11
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 $x^2+9y^2=k^2$ is $\dfrac{m}{n}$, where $m$ and $n$ are coprime, then $2m+n$ is equal to:
Step-by-Step Solution
Key Concept: Centre of circle $=(3/2,1)$ is midpoint of $AB$. Line: $2(3/2)+3(1)-k=0\Rightarrow k=6$. Ellipse: $x^2+9y^2=36\Rightarrow\frac{x^2}{36}+\frac{y^2}{4}=1$. $a^2=36$, $b^2=4$. Latus rectum $=2b^2/a=8/6=4/3$.
$k=6$. Ellipse: $\frac{x^2}{36}+\frac{y^2}{4}=1$. LR $=4/3$. $2(4)+3=11$.
Correct Answer: 1