Conic Sections · JEE Advanced previous year questions

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2=x$, such that the area of the region enclosed between the parabola and the chord is $\frac{4}{3}$. Let $\mathcal{R}$ denote the region lying in the first quadrant, enclosed by the parabola $y^2=x$, the curve $S$, and the lines $x=1$ and $x=4$. Then which of the following statements is (are) TRUE?

1. A.$(4,\sqrt{3})\in S$
2. B.$(5,\sqrt{2})\in S$
3. C.Area of $\mathcal{R}$ is $\frac{14}{3}-2\sqrt{3}$
4. D.Area of $\mathcal{R}$ is $\frac{14}{3}-\sqrt{3}$

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.$q=1$, $p=6\sqrt{3}$

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 ____.

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 $(-6,0)$

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.$(3,4)$