Conic Sections · JEE Advanced previous year questions

2024 2 questions

1. Q42024 · 26 May 2024 · Paper 1

   Consider the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let $S(p,q)$ be a point in the first quadrant such that $\frac{p^2}{9}+\frac{q^2}{4}>1$. Two tangents are drawn from $S$ to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point $T$ in the fourth quadrant. Let $R$ be the vertex of the ellipse with positive $x$-coordinate and $O$ be the center of the ellipse. If the area of the triangle $\Delta ORT$ is $\frac{3}{2}$, then which of the following options is correct?

   1. A.$q=2$, $p=3\sqrt{3}$
   2. B.$q=2$, $p=4\sqrt{3}$
   3. C.$q=1$, $p=5\sqrt{3}$
   4. D.$q=1$, $p=6\sqrt{3}$
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   Answer: A

2. Q122024 · 26 May 2024 · Paper 2· Numerical answer

   A normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0,-\alpha )$ to the parabola $x^2=-4ay$, where $a>0$. Let $L$ be the line passing through $(0,-\alpha )$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r:s=1:16$, then the value of $24a$ is ____.

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   Answer: 12