Conic Sections · JEE Main previous year questions

The equation of a tangent to the parabola, $x^2=8y$, which makes an angle $\theta$ with the positive directions of x-axis, is :

1. A.$x=y\cot \theta -2\tan \theta$
2. B.$y=x\tan \theta +2\cot \theta$
3. C.$x=y\cot \theta +2\tan \theta$
4. D.$y=x\tan \theta -2\cot \theta$

The area (in sq. units) in the first quadrant bounded by the parabola, $y=x^2+1$, the tangent to it at the point $(2,5)$ and the coordinate axes is :

1. A.$\frac{8}{3}$
2. B.$\frac{14}{3}$
3. C.$\frac{187}{24}$
4. D.$\frac{37}{24}$

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

1. A.$(4\sqrt{2},2\sqrt{3})$
2. B.$(4\sqrt{3},2\sqrt{3})$
3. C.$(4\sqrt{3},2\sqrt{2})$
4. D.$(4\sqrt{2},2\sqrt{2})$

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is :

1. A.$\frac{13}{6}$
2. B.2
3. C.$\frac{13}{12}$
4. D.$\frac{13}{8}$

If the area of the triangle whose one vertex is at the vertex of the parabola, $y^2+4(x-a^2)=0$ and the othertwo vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is :

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

Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x-axis then which of the following points does not lie on it?

1. A.$(5,2\sqrt{6})$
2. B.$(6,4\sqrt{2})$
3. C.$(8,6)$
4. D.$(4,-4)$

Let $0<\theta <\frac{\pi }{2}$. If the eccentricity of the hyperbola $\frac{x^2}{{\cos }^2\theta }-\frac{y^2}{{\sin }^2\theta }=1$ is greater than 2, then the length of its latus rectum lies in the interval :

1. A.$(3,\infty )$
2. B.$\left(\frac{3}{2},2\right]$
3. C.$\left(1,\frac{3}{2}\right]$
4. D.$(2,3]$

The area (in sq. units) bounded by the parabola $y=x^2-1$, the tangent at the point $(2,3)$ to it and the y-axis is :

1. A.$\frac{56}{3}$
2. B.$\frac{32}{3}$
3. C.$\frac{8}{3}$
4. D.$\frac{14}{3}$

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is $\frac{3}{2}$ units, then its eccentricity is :

1. A.$\frac{1}{2}$
2. B.$\frac{1}{3}$
3. C.$\frac{2}{3}$
4. D.$\frac{1}{9}$

The differential equation representing the family of ellipse having foci either on the x-axis or on the y-axis, center at the origin and passing through the point (0, 3) is :

1. A.$xy{y}^{\prime \prime }+x(y^{\prime }{)}^2-y{y}^{\prime }=0$
2. B.$x+y{y}^{\prime \prime }=0$
3. C.$xy{y}^{\prime }+y^2-9=0$
4. D.$xy{y}^{\prime }-y^2+9=0$

If the tangents drawn to the hyperbola $4y^2=x^2+1$ intersect the co-ordinate axes at the distinct points A and B then the locus of the mid point of AB is :

1. A.$x^2-4y^2+16x^2{y}^2=0$
2. B.$x^2-4y^2-16x^2{y}^2=0$
3. C.$4x^2-y^2+16x^2{y}^2=0$
4. D.$4x^2-y^2-16x^2{y}^2=0$

Two parabolas with a common vertex and with axes along x-axis and y-axis, respectively intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is :

1. A.$4(x+y)+3=0$
2. B.$3(x+y)+4=0$
3. C.$8(2x+y)+3=0$
4. D.$x+2y+3=0$

If $\beta$ is one of the angles between the normals to the ellipse, $x^2+3y^2=9$ at the points $(3\cos \theta ,\sqrt{3}\sin \theta )$ and $(-3\sin \theta ,\sqrt{3}\cos \theta )$; $\theta \in \left(0,\frac{\pi }{2}\right)$; then $\frac{2\cot \beta }{\sin 2\theta }$ is equal to :

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

Tangents are drawn to the hyperbola $4x^2-y^2=36$ at the points P and Q. If these tangents intersect at the point T(0, 3) then the area (in sq. units) of $\Delta \mathrm{PTQ}$ is :

1. A.$36\sqrt{5}$
2. B.$45\sqrt{5}$
3. C.$54\sqrt{3}$
4. D.$60\sqrt{3}$

The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points $(4,-1)$ and $(-2,2)$ is :

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

If $y=mx+c$ is the normal at a point on the parabola $y^2=8x$ whose focal distance is 8 units, then $|c|$ is equal to :

1. A.$2\sqrt{3}$
2. B.$8\sqrt{3}$
3. C.$10\sqrt{3}$
4. D.$16\sqrt{3}$

Consider an ellipse, whose center is at the origin and its major axis is along the x-axis. If its eccentricity is $\frac{3}{5}$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilatateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is :

1. A.$8$
2. B.$32$
3. C.$80$
4. D.$40$

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is :

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

Let $O$ be the vertex and $Q$ be any point on the parabola, $x^2=8y$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1:3$, then locus of $P$ is :

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

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$, is :

1. A.$\frac{27}{2}$
2. B.$27$
3. C.$\frac{27}{4}$
4. D.$18$

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2+3y^2=6$ on any tangent to it is :

1. A.$(x^2+y^2{)}^2=6x^2+2y^2$
2. B.$(x^2+y^2{)}^2=6x^2-2y^2$
3. C.$(x^2-y^2{)}^2=6x^2+2y^2$
4. D.$(x^2-y^2{)}^2=6x^2-2y^2$

The slope of the line touching both the parabolas $y^2=4x$ and $x^2=-32y$ is

1. A.$\frac{1}{8}$
2. B.$\frac{2}{3}$
3. C.$\frac{1}{2}$
4. D.$\frac{3}{2}$

The area bounded by the parabola $y^2=4x$ and the line $2x-3y+4=0$, in square unit, is

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

The chord $PQ$ of the parabola $y^2=x$, where one end $P$ of the chord is at point $(4,-2)$, is perpendicular to the axis of the parabola. Then the slope of the normal at $Q$ is

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

The normal at $(2,\frac{3}{2})$ to the ellipse, $\frac{x^2}{16}+\frac{y^2}{3}=1$ touches a parabola, whose equation is

1. A.$y^2=-104x$
2. B.$y^2=14x$
3. C.$y^2=26x$
4. D.$y^2=-14x$