Conic Sections · JEE Advanced previous year questions

2023 2 questions

1. Q22023 · 4 Jun 2023 · Paper 1· One or more correct

   Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E:\frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P:y^2=12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?

   1. A.The area of the quadrilateral $A_1{A}_2{A}_3{A}_4$ is 35 square units
   2. B.The area of the quadrilateral $A_1{A}_2{A}_3{A}_4$ is 36 square units
   3. C.The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-3,0)$
   4. D.The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-6,0)$
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   Answer: A, C

2. Q72023 · 4 Jun 2023 · Paper 1

   Let $P$ be a point on the parabola $y^2=4ax$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $PFQ$, where $F$ is the focus of the parabola, is 120. If the slope $m$ of the normal and $a$ are both positive integers, then the pair $(a,m)$ is

   1. A.$(2,3)$
   2. B.$(1,3)$
   3. C.$(2,4)$
   4. D.$(3,4)$
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   Answer: A