Conic Sections · JEE Main previous year questions

If a hyperbola passes through the point P(10, 16) and it has vertices at (± 6, 0), then the equation of the normal to it at P is :

1. A.2x + 5y = 100
2. B.x + 3y = 58
3. C.x + 2y = 42
4. D.3x + 4y = 94

Let a line $y=mx$ ($m>0$) intersect the parabola, $y^2=x$ at a point P, other than the origin. Let the tangent to it at P meet the x-axis at the point Q. If area $(\Delta \mathrm{OPQ})=4\text{ sq. units}$, then m is equal to _______.

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is :

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

The area of the region, enclosed by the circle $x^2+y^2=2$ which is not common to the region bounded by the parabola $y^2=x$ and the straight line $y=x$, is:

1. A.$\frac{1}{6}(24\pi -1)$
2. B.$\frac{1}{3}(12\pi -1)$
3. C.$\frac{1}{3}(6\pi -1)$
4. D.$\frac{1}{6}(12\pi -1)$

If $y=mx+4$ is a tangent to both the parabolas, $y^2=4x$ and $x^2=2by$, then b is equal to :

1. A.-128
2. B.128
3. C.-64
4. D.-32

If $3x+4y=12\sqrt{2}$ is a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{9}=1$ for some $a\in \mathbb{R}$, then the distance between the foci of the ellipse is :

1. A.$2\sqrt{5}$
2. B.$2\sqrt{7}$
3. C.4
4. D.$2\sqrt{2}$

Let P be the point of intersection of the common tangents to the parabola $y^2=12x$ and the hyperbola $8x^2-y^2=8$. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio :

1. A.$14:13$
2. B.$13:11$
3. C.$5:4$
4. D.$2:1$

If the normal to the ellipse $3x^2+4y^2=12$ at a point P on it is parallel to the line, $2x+y=4$ and the tangent to the ellipse at P passes through Q(4,4) then PQ is equal to :

1. A.$\frac{\sqrt{61}}{2}$
2. B.$\frac{\sqrt{221}}{2}$
3. C.$\frac{\sqrt{157}}{2}$
4. D.$\frac{5\sqrt{5}}{2}$

If the area (in sq. units) bounded by the parabola $y^2=4\lambda x$ and the line $y=\lambda x$, $\lambda >0$, is $\frac{1}{9}$ , then $\lambda$ is equal to :

1. A.$4\sqrt{3}$
2. B.$2\sqrt{6}$
3. C.48
4. D.24

An ellipse, with foci at $(0,2)$ and $(0,-2)$ and minor axis of length 4, passes through which of the following points?

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

If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2\sqrt{3})$ is $5x=4\sqrt{5}$ and its eccentricity is e, then :

1. A.$4e^4-24e^2+27=0$
2. B.$4e^4-24e^2+35=0$
3. C.$4e^4-12e^2-27=0$
4. D.$4e^4+8e^2-35=0$

If the line $x-2y=12$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $\left(3,-\frac{9}{2}\right)$ , then the length of the latus rectum of the ellipse is :

1. A.5
2. B.9
3. C.$8\sqrt{3}$
4. D.$12\sqrt{2}$

If $5x+9=0$ is the directrix of the hyperbola $16x^2-9y^2=144$, then its corresponding focus is :

1. A.$\left(\frac{5}{3},0\right)$
2. B.(5, 0)
3. C.(- 5, 0)
4. D.$\left(-\frac{5}{3},0\right)$

The tangent and normal to the ellipse $3x^2+5y^2=32$ at the point P(2, 2) meet the x-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is :

1. A.$\frac{14}{3}$
2. B.$\frac{16}{3}$
3. C.$\frac{68}{15}$
4. D.$\frac{34}{15}$

If one end of a focal chord of the parabola, $y^2=16x$ is at (1, 4), then the length of this focal chord is :

1. A.24
2. B.20
3. C.25
4. D.22

If the line $y=mx+7\sqrt{3}$ is normal to the hyperbola $\frac{x^2}{24}-\frac{y^2}{18}=1$, then a value of m is :

1. A.$\frac{3}{\sqrt{5}}$
2. B.$\frac{\sqrt{15}}{2}$
3. C.$\frac{\sqrt{5}}{2}$
4. D.$\frac{2}{\sqrt{5}}$

If the tangent to the parabola $y^2=x$ at a point $(\alpha ,\beta )$, $(\beta >0)$ is also a tangent to the ellipse, $x^2+2y^2=1$, then $\alpha$ is equal to :

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

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $(0,5\sqrt{3})$, then the length of its latus rectum is :

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

If the eccentricity of the standard hyperbola passing through the point (4,6) is 2, then the equation of the tangent to the hyperbola at (4,6) is :

1. A.2x – y – 2 = 0
2. B.3x – 2y = 0
3. C.2x – 3y + 10 = 0
4. D.x – 2y + 8 = 0

The tangent to the parabola $y^2=4x$ at the point where it intersects the circle $x^2+y^2=5$ in the first quadrant, passes through the point :

1. A.$\left(-\frac{1}{4},\frac{1}{2}\right)$
2. B.$\left(-\frac{1}{3},\frac{4}{3}\right)$
3. C.$\left(\frac{3}{4},\frac{7}{4}\right)$
4. D.$\left(\frac{1}{4},\frac{3}{4}\right)$

The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, $y=12-x^2$ such that the rectangle lies inside the parabola, is :

1. A.36
2. B.$20\sqrt{2}$
3. C.$18\sqrt{3}$
4. D.32

If the vertices of a hyperbola be at $(-2,0)$ and $(2,0)$ and one of its foci be at $(-3,0)$, then which one of the following points does not lie on this hyperbola?

1. A.$(6,5\sqrt{2})$
2. B.$(2\sqrt{6},5)$
3. C.$(-6,2\sqrt{10})$
4. D.$(4,\sqrt{15})$

The area (in sq. units) of the region bounded by the parabola, $y=x^2+2$ and the lines, $y=x+1$, $x=0$ and $x=3$, is

1. A.$\frac{15}{4}$
2. B.$\frac{15}{2}$
3. C.$\frac{21}{2}$
4. D.$\frac{17}{4}$

Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola, $y^2=4x$ and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $\Delta \mathrm{PXQ}$ is maximum. Then this maximum area (in sq. units) is :

1. A.$\frac{625}{4}$
2. B.$\frac{125}{4}$
3. C.$\frac{75}{2}$
4. D.$\frac{125}{2}$

Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $\Delta {\mathrm{S}}^{\prime }\mathrm{BS}$ is a right angled triangle with right angle at B and area $(\Delta {\mathrm{S}}^{\prime }\mathrm{BS})=8sq.\mathrm{units}$, then the length of a latus rectum of the ellipse is :

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