Conic Sections · JEE Main previous year questions

If a tangent to the ellipse $x^2+4y^2=4$ meets the tangents at the extremities of it major axis at B and C, then the circle with BC as diameter passes through the point :

1. A.$(\sqrt{3},0)$
2. B.$(\sqrt{2},0)$
3. C.(1, 1)
4. D.$(-1,1)$

Let $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$. Let $E_2$ be another ellipse such that it touches the end points of major axis of $E_1$ and the foci of $E_2$ are the end points of minor axis of $E_1$. If $E_1$ and $E_2$ have same eccentricities, then its value is :

1. A.$\frac{-1+\sqrt{5}}{2}$
2. B.$\frac{-1+\sqrt{8}}{2}$
3. C.$\frac{-1+\sqrt{3}}{2}$
4. D.$\frac{-1+\sqrt{6}}{2}$

Let a line $L:2x+y=k$, $k>0$ be a tangent to the hyperbola $x^2-y^2=3$. If L is also a tangent to the parabola $y^2=\alpha x$, then $\alpha$ is equal to :

1. A.$12$
2. B.$-12$
3. C.$24$
4. D.$-24$

Let the tangent to the parabola $S:y^2=2x$ at the point $P(2,2)$ meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to :

1. A.$\frac{25}{2}$
2. B.$\frac{35}{2}$
3. C.$\frac{15}{2}$
4. D.$25$

Let T be the tangent to the ellipse $E:x^2+4y^2=5$ at the point $P(1,1)$. If the area of the region bounded by the tangent T, ellipse E, lines $x=1$ and $x=\sqrt{5}$ is $\alpha \sqrt{5}+\beta +\gamma {\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)$, then $|\alpha +\beta +\gamma |$ is equal to ________.

Consider a hyperbola $\mathrm{H}:x^2-2y^2=4$. Let the tangent at a point $\mathrm{P}(4,\sqrt{6})$ meet the x-axis at Q and latus rectum at $\mathrm{R}(x_1,y_1),x_1>0$. If F is a focus of H which is nearer to the point P, then the area of $\Delta \mathrm{QFR}$ is equal to :

1. A.$\sqrt{6}-1$
2. B.$\frac{7}{\sqrt{6}}-2$
3. C.$4\sqrt{6}-1$
4. D.$4\sqrt{6}$

Let a tangent be drawn to the ellipse $\frac{x^2}{27}+y^2=1$ at $(3\sqrt{3}\cos \theta ,\sin \theta )$ where $\theta \in \left(0,\frac{\pi }{2}\right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

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

If the three normals drawn to the parabola, $y^2=2x$ pass through the point $(a,0)$, $a\ne 0$, then 'a' must be greater than :

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

A hyperbola passes through the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is :

1. A.$\frac{x^2}{9}-\frac{y^2}{4}=1$
2. B.$\frac{x^2}{9}-\frac{y^2}{16}=1$
3. C.$\frac{x^2}{9}-\frac{y^2}{25}=1$
4. D.$x^2-y^2=9$

Let $L_1$ be a tangent to the parabola $y^2=4(x+1)$ and $L_2$ be a tangent to the parabola $y^2=8(x+2)$ such that $L_1$ and $L_2$ intersect at right angles. Then $L_1$ and $L_2$ meet on the straight line :

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

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $$\frac{x^2}{4}+\frac{y^2}{2}=1$$ from any of its foci?

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

If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies :

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

A hyperbola having the transverse axis of length $\sqrt{2}$ has the same foci as that of the ellipse $3x^2+4y^2=12$, then this hyperbola does not pass through which of the following points?

1. A.$\left(1,-\frac{1}{\sqrt{2}}\right)$
2. B.$\left(\sqrt{\frac{3}{2}},\frac{1}{2}\right)$
3. C.$\left(-\sqrt{\frac{3}{2}},1\right)$
4. D.$\left(\frac{1}{\sqrt{2}},0\right)$

Let P be a point on the parabola, $y^2=12x$ and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the y-intercept of the line NQ is $\frac{4}{3}$, then :

1. A.$\mathrm{MQ}=\frac{1}{3}$
2. B.$\mathrm{PN}=4$
3. C.$\mathrm{PN}=3$
4. D.$\mathrm{MQ}=\frac{1}{4}$

Let $e_1$ and $e_2$ be the eccentricities of the ellipse, $\frac{x^2}{25}+\frac{y^2}{b^2}=1(b<5)$ and the hyperbola, $\frac{x^2}{16}-\frac{y^2}{b^2}=1$ respectively satisfying $e_1{e}_2=1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $(\alpha ,\beta )$ is equal to :

1. A.$(8,10)$
2. B.$(8,12)$
3. C.$\left(\frac{24}{5},10\right)$
4. D.$\left(\frac{20}{3},12\right)$

If the tangent to the curve, $y=e^x$ at a point $(c,e^c)$ and the normal to the parabola, $y^2=4x$ at the point $(1,2)$ intersect at the same point on the x-axis, then the value of c is _______.

Area (in sq. units) of the region outside $\frac{|x|}{2}+\frac{|y|}{3}=1$ and inside the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is :

1. A.$6(4-\pi )$
2. B.$3(4-\pi )$
3. C.$6(\pi -2)$
4. D.$3(\pi -2)$

A line parallel to the straight line $2x-y=0$ is tangent to the hyperbola $\frac{x^2}{4}-\frac{y^2}{2}=1$ at the point $(x_1,y_1)$. Then $x_1^2+5y_1^2$ is equal to :

1. A.5
2. B.6
3. C.10
4. D.8

The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^2=8x$, with one of its vertices on the vertex of this parabola, is :

1. A.$256\sqrt{3}$
2. B.$64\sqrt{3}$
3. C.$128\sqrt{3}$
4. D.$192\sqrt{3}$

For some $\theta \in \left(0,\frac{\pi }{2}\right)$, if the eccentricity of the hyperbola, $x^2-y^2{\sec }^2\theta =10$ is $\sqrt{5}$ times the eccentricity of the ellipse, $x^2{\sec }^2\theta +y^2=5$, then the length of the latus rectum of the ellipse, is :

1. A.$\sqrt{30}$
2. B.$2\sqrt{6}$
3. C.$\frac{4\sqrt{5}}{3}$
4. D.$\frac{2\sqrt{5}}{3}$

If $e_1$ and $e_2$ are the eccentricities of the ellipse, $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola, $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and $(e_1,e_2)$ is a point on the ellipse, $15x^2+3y^2=k$, then $k$ is equal to :

1. A.17
2. B.16
3. C.15
4. D.14

The length of the minor axis (along y-axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line, $x+6y=8$; then its eccentricity is :

1. A.$\frac{1}{3}\sqrt{\frac{11}{3}}$
2. B.$\frac{1}{2}\sqrt{\frac{5}{3}}$
3. C.$\sqrt{\frac{5}{6}}$
4. D.$\frac{1}{2}\sqrt{\frac{11}{3}}$

If one end of a focal chord AB of the parabola $y^2=8x$ is at $A\left(\frac{1}{2},-2\right)$, then the equation of the tangent to it at B is :

1. A.$2x-y-24=0$
2. B.$x-2y+8=0$
3. C.$x+2y+8=0$
4. D.$2x+y-24=0$

The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola, $x^2=4y$, internally in the ratio $1:2$, is :

1. A.$9x^2-3y=2$
2. B.$4x^2-3y=2$
3. C.$x^2-3y=2$
4. D.$9x^2-12y=8$

Let the line $y=mx$ and the ellipse $2x^2+y^2=1$ intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at $\left(-\frac{1}{3\sqrt{2}},0\right)$ and $(0,\beta )$, then $\beta$ is equal to :

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