Conic Sections · JEE Main previous year questions

Let PQ be a chord of the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$, perpendicular to the x -axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt{3}$, then the area of the triangle OPQ is

1. A.$2\sqrt{3}$
2. B.$\frac{11}{5}$
3. C.$\frac{8\sqrt{3}}{5}$
4. D.$\frac{9}{5}$

If the line $\alpha x+2y=1$, where $\alpha \in \mathbb{R}$, does not meet the hyperbola $x^2-9y^2=9$, then a possible value of $\alpha$ is :

1. A.0.6
2. B.0.7
3. C.0.8
4. D.0.5

If the chord joining the points ${\mathrm{P}}_1(x_1,y_1)$ and ${\mathrm{P}}_2(x_2,y_2)$ on the parabola $y^2=12x$ subtends a right angle at the vertex of the parabola, then $x_1{x}_2-y_1{y}_2$ is equal to

1. A.280
2. B.288
3. C.292
4. D.284

The area of the region, inside the ellipse $x^2+4y^2=4$ and outside the region bounded by the curves $y=|x|-1$ and $y=1-|x|$, is :

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

Let the foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$. If the eccentricity of the hyperbola is 5 , then the length of its latus rectum is :

1. A.$\frac{96}{\sqrt{5}}$
2. B.$24\sqrt{5}$
3. C.12
4. D.16

Let O be the vertex of the parabola $x^2=4y$ and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio $2:3$ be the conic $C$. Then the equation of the chord of $C$, which is bisected at the point $(1,2)$, is :

1. A.$5x-4y+3=0$
2. B.$x-2y+3=0$
3. C.$4x-5y+6=0$
4. D.$5x-y-3=0$

If the line $\alpha x+4y=\sqrt{7}$, where $\alpha \in \mathbf{R}$, touches the ellipse $3x^2+4y^2=1$ at the point P in the first quadrant, then one of the focal distances of $P$ is :

1. A.$\frac{1}{\sqrt{3}}-\frac{1}{2\sqrt{11}}$
2. B.$\frac{1}{\sqrt{3}}-\frac{1}{2\sqrt{5}}$
3. C.$\frac{1}{\sqrt{3}}+\frac{1}{2\sqrt{5}}$
4. D.$\frac{1}{\sqrt{3}}+\frac{1}{2\sqrt{7}}$

Let $y^2=12x$ be the parabola with its vertex at $O$. Let $P$ be a point on the parabola and $A$ be a point on the $x$-axis such that $\angle OPA=90^{\circ }$. Then the locus of the centroid of such triangles $OPA$ is:

1. A.$y^2-4x+8=0$
2. B.$y^2-2x+8=0$
3. C.$y^2-9x+6=0$
4. D.$y^2-6x+4=0$

Let one end of a focal chord of the parabola $y^2=16x$ be $(16,16)$. If $P(\alpha ,\beta )$ divides this focal chord internally in the ratio $5:2$, then the minimum value of $\alpha +\beta$ is equal to:

1. A.5
2. B.7
3. C.22
4. D.16

The length of the latus-rectum of the ellipse, whose foci are $(2,5)$ and $(2,-3)$ and eccentricity is $\frac{4}{5}$, is

1. A.$\frac{50}{3}$
2. B.$\frac{18}{5}$
3. C.$\frac{6}{5}$
4. D.$\frac{10}{3}$

Let the sum of the focal distances of the point $\mathrm{P}(4,3)$ on the hyperbola $\mathrm{H}:\frac{x^2}{{\mathrm{a}}^2}-\frac{y^2}{{\mathrm{b}}^2}=1$ be $8\sqrt{\frac{5}{3}}$. If for H , the length of the latus rectum is $l$ and the product of the focal distances of the point P is m , then $9l^2+6\mathrm{m}$ is equal to :

1. A.187
2. B.184
3. C.186
4. D.185

Let for two distinct values of p the lines $y=x+p$ touch the ellipse $\mathrm{E}:\frac{x^2}{4^2}+\frac{y^2}{3^2}=1$ at the points A and B . Let the line $y=x$ intersect E at the points C and D . Then the area of the quadrilateral $ABCD$ is equal to :

1. A.48
2. B.20
3. C.24
4. D.36

The centre of a circle C is at the centre of the ellipse $\mathrm{E}:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$. Let C pass through the foci $F_1$ and $F_2$ of E such that the circle $C$ and the ellipse $E$ intersect at four points. Let P be one of these four points. If the area of the triangle ${\mathrm{PF}}_1{\mathrm{F}}_2$ is 30 and the length of the major axis of $E$ is 17 , then the distance between the foci of $E$ is :

1. A.12
2. B.26
3. C.13
4. D.$\frac{13}{2}$

A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $\mathrm{P}:y^2=x-2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line $AB$, parabola $P$ and the $x$-axis, is :

1. A.3
2. B.$\frac{7}{3}$
3. C.$\frac{8}{3}$
4. D.2

The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2\sqrt{2}$ units from the origin, respectively. If the point $(1,k)$ lies on the parabola, then a possible value of $k$ is :

1. A.8
2. B.3
3. C.9
4. D.4

Let the values of p , for which the shortest distance between the lines $\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}$ and $\vec{r}=(p\hat{i}+2\hat{j}+\hat{k})+\lambda (2\hat{i}+3\hat{j}+4\hat{k})$ is $\frac{1}{\sqrt{6}}$, be a, b, $(a<b)$. Then the length of the latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is :

1. A.$\frac{3}{2}$
2. B.9
3. C.18
4. D.$\frac{2}{3}$

A line passing through the point $P(\sqrt{5},\sqrt{5})$ intersects the ellipse $\frac{x^2}{36}+\frac{y^2}{25}=1$ at $A$ and $B$ such that $(PA)\cdot (PB)$ is maximum. Then $5\left(P{A}^2+P{B}^2\right)$ is equal to :

1. A.290
2. B.377
3. C.338
4. D.218

Let the product of the focal distances of the point $\mathbf{P}(4,2\sqrt{3})$ on the hyperbola $\mathrm{H}:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be 32 . Let the length of the conjugate axis of H be $p$ and the length of its latus rectum be $q$. Then $p^2+q^2$ is equal to____

Let $C$ be the circle of minimum area enclosing the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{1}{2}$ and foci $(\pm 2,0)$. Let $PQR$ be a variable triangle, whose vertex $P$ is on the circle $C$ and the side $QR$ of length $2a$ is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$-axis. Then the maximum area of the triangle $PQR$ is :

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

If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3x^2-y^2-\alpha x+\beta y+\gamma =0$, then $\alpha +\beta +\gamma$ is equal to_________.

If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and $P$ be a point on the ellipse, then $\min (SP\cdot S^{\prime }P)+\max (SP\cdot S^{\prime }P)$ is equal to :

1. A.$3(6+\sqrt{2})$
2. B.$3(1+\sqrt{2})$
3. C.27
4. D.9

Let the focal chord $PQ$ of the parabola $y^2=4x$ make an angle of $60^{\circ }$ with the positive $x$ axis, where $P$ lies in the first quadrant. If the circle, whose one diameter is $PS$, $S$ being the focus of the parabola, touches the $y$-axis at the point $(0,\alpha )$, then $5{\alpha }^2$ is equal to:

1. A.15
2. B.25
3. C.20
4. D.30

Let one focus of the hyperbola $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be at $(\sqrt{10},0)$ and the corresponding directrix be $x=\frac{9}{\sqrt{10}}$. If $e$ and $l$ respectively are the eccentricity and the length of the latus rectum of $H$, then $9(e^2+l)$ is equal to :

1. A.12
2. B.14
3. C.15
4. D.16

Let the point P of the focal chord PQ of the parabola $y^2=16x$ be $(1,-4)$. If the focus of the parabola divides the chord $PQ$ in the ratio $m:n,\gcd (m,n)=1$, then $m^2+n^2$ is equal to :

1. A.17
2. B.37
3. C.10
4. D.26

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

1. A.$\frac{3}{\sqrt{19}}$
2. B.$\frac{\sqrt{3}}{16}$
3. C.$\frac{4}{\sqrt{17}}$
4. D.$\frac{\sqrt{5}}{7}$