Conic Sections · JEE Main previous year questions

The area of triangle formed by the lines joining the vertex of the parabola, $x^2=8y$, to the extremities of its latus rectum is

1. A.2
2. B.8
3. C.1
4. D.4

If $P_1$ and $P_2$ are two points on the ellipse $\frac{x^2}{4}+y^2=1$ at which the tangents are parallel to the chord joining the points $(0,1)$ and $(2,0)$, then the distance between $P_1$ and $P_2$ is

1. A.$2\sqrt{2}$
2. B.$\sqrt{5}$
3. C.$2\sqrt{3}$
4. D.$\sqrt{10}$

If the eccentricity of a hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$, which passes through $(K,2)$, is $\frac{\sqrt{13}}{3}$, then the value of $K^2$ is

1. A.18
2. B.8
3. C.1
4. D.2

Statement 1: $y=mx-\frac{1}{m}$ is always a tangent to the parabola, $y^2=-4x$ for all non-zero values of $m$. Statement 2: Every tangent to the parabola, $y^2=-4x$ will meet its axis at a point whose abscissa is non-negative.

1. A.Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
2. B.Statement 1 is false, Statement 2 is true.
3. C.Statement 1 is true, Statement 2 is false.
4. D.Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.

If two tangents drawn from a point $P$ to the parabola $y^2=4x$ are at right angles, then the locus of $P$ is

1. A.$2x+1=0$
2. B.$x=-1$
3. C.$2x-1=0$
4. D.$x=1$

The ellipse $x^2+4y^2=4$ is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point $(4,0)$. Then the equation of the ellipse is :

1. A.$x^2+12y^2=16$
2. B.$4x^2+48y^2=48$
3. C.$4x^2+64y^2=48$
4. D.$x^2+16y^2=16$

The area of the region bounded by the parabola $(y-2{)}^2=x-1$, the tangent of the parabola at the point $(2,3)$ and the $x$-axis is :

1. A.6
2. B.9
3. C.12
4. D.3

The locus of the vertices of the family of parabolas $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$ is :

1. A.$xy=\frac{105}{64}$
2. B.$xy=\frac{3}{4}$
3. C.$xy=\frac{35}{16}$
4. D.$xy=\frac{64}{105}$

In the ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is :

1. A.$\frac{3}{5}$
2. B.$\frac{1}{2}$
3. C.$\frac{4}{5}$
4. D.$\frac{1}{\sqrt{5}}$

An ellipse has $OB$ as semi minor axis, $F$ and $F^{\prime }$ its foci and the angle $FB{F}^{\prime }$ is a right angle. Then the eccentricity of the ellipse is :

1. A.$\frac{1}{\sqrt{2}}$
2. B.$\frac{1}{2}$
3. C.$\frac{1}{4}$
4. D.$\frac{1}{\sqrt{3}}$

Let $P$ be the point $(1,0)$ and $Q$ a point on the parabola $y^2=8x$. The locus of mid point of $PQ$ is :

1. A.$y^2-4x+2=0$
2. B.$y^2+4x+2=0$
3. C.$x^2+4y+2=0$
4. D.$x^2-4y+2=0$

Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

1. A.$2ab$
2. B.$ab$
3. C.$\sqrt{ab}$
4. D.$\frac{a}{b}$

The parabolas $y^2=4x$ and $x^2=4y$ divide the square region bounded by the lines $x=4,y=4$ and the coordinate axes. If $S_1,S_2,S_3$ are respectively the areas of these parts numbered from top to bottom ; then $S_1:S_2:S_3$ is :

1. A.$1:2:1$
2. B.$1:2:3$
3. C.$2:1:2$
4. D.$1:1:1$

The locus of a point $P(\alpha ,\beta )$ moving under the condition that the line $y=\alpha x+\beta$ is tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is :

1. A.an ellipse
2. B.a circle
3. C.a parabola
4. D.a hyperbola

If $a\ne 0$ and the line $2bx+3cy+4d=0$ passes through the points of intersection of the parabolas $y^2=4ax$ and $x^2=4ay$, then :

1. A.$d^2+(3b-2c{)}^2=0$
2. B.$d^2+(3b+2c{)}^2=0$
3. C.$d^2+(2b-3c{)}^2=0$
4. D.$d^2+(2b+3c{)}^2=0$

The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}$. If one of the directrices is $x=4$, then the equation of the ellipse is :

1. A.$4x^2+3y^2=1$
2. B.$3x^2+4y^2=12$
3. C.$4x^2+3y^2=12$
4. D.$3x^2+4y^2=1$

A point on the parabola $y^2=18x$ at which the ordinate increases at twice the rate of the abscissa is

1. A.$\left(\frac{9}{8},\frac{9}{2}\right)$
2. B.$(2,-4)$
3. C.$\left(\frac{-9}{8},\frac{9}{2}\right)$
4. D.$(2,4)$