Conic Sections · JEE Main previous year questions

If the maximum distance of normal to the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1,b<2$, from the origin is 1, then the eccentricity of the ellipse is :

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

Let $y=f(x)$ represent a parabola with focus $(-\frac{1}{2},0)$ and directrix $y=-\frac{1}{2}$. Then $S=\left\{x\in \mathbb{R}:{\tan }^{-1}(\sqrt{f(x)})+{\sin }^{-1}(\sqrt{f(x)+1})=\frac{\pi }{2}\right\}:$

1. A.is an empty set
2. B.contains exactly one element
3. C.contains exactly two elements
4. D.is an infinite set

Let $\mathrm{H}$ be the hyperbola, whose foci are $(1\pm \sqrt{2},0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is :

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

Let S be the set of all $a\in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $\mathrm{P}(b,c),b,c\in \mathbb{N}$, on the parabola $y^2=2ax$ and the lines $x=b,y=0$ is $16{\text{ unit}}^2$, then ${\sum }_{a\in \mathrm{S}}a$ is equal to :

If $\mathrm{P}(h,k)$ be a point on the parabola $x=4y^2$, which is nearest to the point $\mathrm{Q}(0,33)$, then the distance of $\mathrm{P}$ from the directrix of the parabola $y^2=4(x+y)$ is equal to :

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

If the tangent at a point P on the parabola $y^2=3x$ is parallel to the line $x+2y=1$ and the tangents at the points Q and R on the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ are perpendicular to the line $x-y=2$, then the area of the triangle PQR is :

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

If the area enclosed by the parabolas ${\mathrm{P}}_1:2y=5x^2$ and ${\mathrm{P}}_2:x^2-y+6=0$ is equal to the area enclosed by ${\mathrm{P}}_1$ and $y=\alpha x,\alpha >0$, then ${\alpha }^3$ is equal to ____.

The vertices of a hyperbola $\mathrm{H}$ are $(\pm 6,0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $\mathrm{N}$ be the normal to $\mathrm{H}$ at a point in the first quadrant and parallel to the line $\sqrt{2}x+y=2\sqrt{2}$. If $\mathrm{d}$ is the length of the line segment of $\mathrm{N}$ between $\mathrm{H}$ and the y-axis then ${\mathrm{d}}^2$ is equal to ____.

The equations of two sides of a variable triangle are $x=0$ and $y=3$, and its third side is a tangent to the parabola $y^2=6x$. The locus of its circumcentre is :

1. A.$4y^2-18y-3x-18=0$
2. B.$4y^2+18y+3x+18=0$
3. C.$4y^2-18y+3x+18=0$
4. D.$4y^2-18y-3x+18=0$

Let T and C respectively be the transverse and conjugate axes of the hyperbola $16x^2-y^2+64x+4y+44=0$. Then the area of the region above the parabola $x^2=y+4$, below the transverse axis T and on the right of the conjugate axis C is :

1. A.$4\sqrt{6}-\frac{28}{3}$
2. B.$4\sqrt{6}-\frac{44}{3}$
3. C.$4\sqrt{6}+\frac{28}{3}$
4. D.$4\sqrt{6}+\frac{44}{3}$

Let a tangent to the curve $y^2=24x$ meet the curve $xy=2$ at the points A and B. Then the mid points of such line segments AB lie on a parabola with the :

1. A.length of latus rectum 2
2. B.directrix $4x=-3$
3. C.directrix $4x=3$
4. D.length of latus rectum $\frac{3}{2}$

The equations of the sides AB and AC of a triangle ABC are $(\lambda +1)x+\lambda y=4$ and $\lambda x+(1-\lambda )y+\lambda =0$ respectively. Its vertex A is on the y-axis and its orthocentre is $(1,2)$. The length of the tangent from the point C to the part of the parabola $y^2=6x$ in the first quadrant is :

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

Let a line $\mathrm{L}$ pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0$, $b\in \mathbf{R}-\left\{\frac{4}{3}\right\}$. If the line $\mathrm{L}$ also passes through the point $(1,1)$ and touches the circle $17\left(x^2+y^2\right)=16$, then the eccentricity of the ellipse $\frac{x^2}{5}+\frac{y^2}{b^2}=1$ is :

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

Let the focal chord of the parabola $\mathrm{P}:y^2=4x$ along the line $\mathrm{L}:y=mx+c$, $m>0$ meet the parabola at the points $\mathrm{M}$ and $\mathrm{N}$. Let the line $\mathrm{L}$ be a tangent to the hyperbola $\mathrm{H}:x^2-y^2=4$. If $\mathrm{O}$ is the vertex of $\mathrm{P}$ and $\mathrm{F}$ is the focus of $\mathrm{H}$ on the positive x-axis, then the area of the quadrilateral $\mathrm{OMFN}$ is :

1. A.$2\sqrt{6}$
2. B.$2\sqrt{14}$
3. C.$4\sqrt{6}$
4. D.$4\sqrt{14}$

If the tangents drawn at the points $\mathrm{P}$ and $\mathrm{Q}$ on the parabola $y^2=2x-3$ intersect at the point $R(0,1)$, then the orthocentre of the triangle $PQR$ is :

1. A.$(0,1)$
2. B.$(2,-1)$
3. C.$(6,3)$
4. D.$(2,1)$

For the hyperbola $\mathrm{H}:x^2-y^2=1$ and the ellipse $\mathrm{E}:\frac{x^2}{{\mathrm{a}}^2}+\frac{y^2}{{\mathrm{b}}^2}=1$, $\mathrm{a}>\mathrm{b}>0$, let the (1) eccentricity of $\mathrm{E}$ be reciprocal of the eccentricity of $\mathrm{H}$, and (2) the line $y=\sqrt{\frac{5}{2}}x+\mathrm{K}$ be a common tangent of $\mathrm{E}$ and $\mathrm{H}$. Then $4\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)$ is equal to ____.

Let the hyperbola $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ pass through the point $(2\sqrt{2},-2\sqrt{2})$. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?

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

Let the tangents at the points P and Q on the ellipse $\frac{x^2}{2}+\frac{y^2}{4}=1$ meet at the point $R(\sqrt{2},2\sqrt{2}-2)$. If S is the focus of the ellipse on its negative major axis, then ${\text{SP}}^2+{\text{SQ}}^2$ is equal to _________.

An ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $H:\frac{x^2}{49}-\frac{y^2}{64}=-1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$, respectively. Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113l$ is equal to _________.

If the length of the latus rectum of the ellipse $x^2+4y^2+2x+8y-\lambda =0$ is 4 , and $l$ is the length of its major axis, then $\lambda +l$ is equal to _________.

If the length of the latus rectum of a parabola, whose focus is $(a,a)$ and the tangent at its vertex is $x+y=a$, is $16$, then $|a|$ is equal to :

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

Let the tangent drawn to the parabola $y^2=24x$ at the point $(\alpha ,\beta )$ is perpendicular to the line $2x+2y=5$. Then the normal to the hyperbola $\frac{x^2}{{\alpha }^2}-\frac{y^2}{{\beta }^2}=1$ at the point $(\alpha +4,\beta +4)$ does NOT pass through the point :

1. A.$(25,10)$
2. B.$(20,12)$
3. C.$(30,8)$
4. D.$(15,13)$

If the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the line $\frac{x}{7}+\frac{y}{2\sqrt{6}}=1$ on the $x$-axis and the line $\frac{x}{7}-\frac{y}{2\sqrt{6}}=1$ on the $y$-axis, then the eccentricity of the ellipse is :

1. A.$\frac{5}{7}$
2. B.$\frac{2\sqrt{6}}{7}$
3. C.$\frac{3}{7}$
4. D.$\frac{2\sqrt{5}}{7}$

Let the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{\alpha }=\frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is :

1. A.$\frac{32}{9}$
2. B.$\frac{18}{5}$
3. C.$\frac{27}{4}$
4. D.$\frac{27}{10}$

The tangents at the points $A(1,3)$ and $B(1,-1)$ on the parabola $y^2-2x-2y=1$ meet at the point $P$. Then the area (in ${\text{unit}}^2$) of the triangle $PAB$ is :

1. A.4
2. B.6
3. C.7
4. D.8