Conic Sections · JEE Main previous year questions

The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2y^2=kx$ and $k{y}^2=2(y-x)$ is maximum, is equal to :

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2},\frac{14}{3}\sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to

1. A.$14\sqrt{\frac{2}{5}}-\frac{4}{3}$
2. B.$7\sqrt{\frac{2}{5}}+\frac{8}{3}$
3. C.$7\sqrt{\frac{2}{5}}-\frac{8}{3}$
4. D.$14\sqrt{\frac{2}{5}}-\frac{16}{3}$

Let the foci and length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$ be $(\pm 5,0)$ and $\sqrt{50}$, respectively. Then, the square of the eccentricity of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{a^2{b}^2}=1$ equals

Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$, of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then, the square of the length of the latus rectum of $E$, is

1. A.$\frac{512}{25}$
2. B.$\frac{656}{25}$
3. C.$\frac{385}{8}$
4. D.$\frac{347}{8}$

The area of the region enclosed by the parabolas $y=4x-x^2$ and $3y=(x-4{)}^2$ is equal to :

1. A.$\frac{32}{9}$
2. B.$\frac{14}{3}$
3. C.4
4. D.6

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

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

The area (in square units) of the region bounded by the parabola $y^2=4(x-2)$ and the line $y=2x-8$, is :

1. A.7
2. B.8
3. C.9
4. D.6

Let the latus rectum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi }{3}$ at the centre of the hyperbola. If $b^2$ is equal to $\frac{l}{m}(1+\sqrt{n})$, where $l$ and $m$ are co-prime numbers, then $l^2+m^2+n^2$ is equal to _________.

Let $P(\alpha ,\beta )$ be a point on the parabola $y^2=4x$. If $P$ also lies on the chord of the parabola $x^2=8y$ whose mid point is $\left(1,\frac{5}{4}\right)$, then $(\alpha -28)(\beta -8)$ is equal to _________.

The length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid point is $\left(1,\frac{2}{5}\right)$, is equal to :

1. A.$\frac{\sqrt{1691}}{5}$
2. B.$\frac{\sqrt{2009}}{5}$
3. C.$\frac{\sqrt{1541}}{5}$
4. D.$\frac{\sqrt{1741}}{5}$

Let $e_1$ be the eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$, which passes through the foci of the hyperbola. If $e_1{e}_2=1$, then the length of the chord of the ellipse parallel to the $x$-axis and passing through $(0,2)$ is :

1. A.$\frac{8\sqrt{5}}{3}$
2. B.$3\sqrt{5}$
3. C.$4\sqrt{5}$
4. D.$\frac{10\sqrt{5}}{3}$

Let $\mathrm{PQ}$ be a focal chord of the parabola $y^2=36x$ of length $100$, making an acute angle with the positive $x$-axis. Let the ordinate of $\mathrm{P}$ be positive and $\mathrm{M}$ be the point on the line segment $\mathrm{PQ}$ such that $\mathrm{PM}:\mathrm{MQ}=3:1$. Then which of the following points does NOT lie on the line passing through $\mathrm{M}$ and perpendicular to the line $\mathrm{PQ}$?

1. A.$(6,29)$
2. B.$(-3,43)$
3. C.$(3,33)$
4. D.$(-6,45)$

Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $\mathrm{P}(4,1)$ to the hyperbola $H:\frac{y^2}{25}-\frac{x^2}{16}=1$. If Q is the point from which the tangents drawn to H have slopes $|m_1|$ and $|m_2|$ and they make positive intercepts $\alpha$ and $\beta$ on the $x$-axis, then $\frac{(PQ{)}^2}{\alpha \beta }$ is equal to _________.

The foci of a hyperbola are $(\pm 2,0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line $2x+3y=6$, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the $x$ - and $y$-axes are $a$ and $b$ respectively, then $|6a|+|5b|$ is equal to ___________

Let $\mathrm{P}\left(\frac{2\sqrt{3}}{\sqrt{7}},\frac{6}{\sqrt{7}}\right)$, $\mathrm{Q}$, $\mathrm{R}$ and $\mathrm{S}$ be four points on the ellipse $9x^2+4y^2=36$. Let $\mathrm{PQ}$ and $\mathrm{RS}$ be mutually perpendicular and pass through the origin. If $\frac{1}{(\mathrm{PQ}{)}^2}+\frac{1}{(\mathrm{RS}{)}^2}=\frac{p}{q}$, where $p$ and $q$ are coprime, then $p+q$ is equal to :

1. A.$143$
2. B.$147$
3. C.$137$
4. D.$157$

Let ${\text{H}}_n:\frac{x^2}{1+n}-\frac{y^2}{3+n}=1,n\in \mathbb{N}$. Let k be the smallest even value of n such that the eccentricity of ${\text{H}}_k$ is a rational number. If $l$ is the length of the latus rectum of ${\text{H}}_k$, then $21l$ is equal to _________.

Let the tangent to the parabola $y^2=12x$ at the point $(3,\alpha )$ be perpendicular to the line $2x+2y=3$. Then the square of distance of the point $(6,-4)$ from the normal to the hyperbola ${\alpha }^2{x}^2-9y^2=9{\alpha }^2$ at its point $(\alpha -1,\alpha +2)$ is equal to _________.

Let the ellipse $E:x^2+9y^2=9$ intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is $\frac{m}{n}$, where m and n are coprime, then $m-n$ is equal to :

1. A.15
2. B.16
3. C.17
4. D.18

Let $y=p(x)$ be the parabola passing through the points $(-1,0)$, $(0,1)$ and $(1,0)$. If the area of the region $\{(x,y):(x+1{)}^2+(y-1{)}^2\le 1,y\le p(x)\}$ is A, then $12(\pi -4A)$ is equal to _________.

Let $R$ be the focus of the parabola $y^2=20x$ and the line $y=mx+c$ intersect the parabola at two points $P$ and $Q$. Let the point $G(10,10)$ be the centroid of the triangle $PQR$. If $c-m=6$, then $(PQ{)}^2$ is :

1. A.$317$
2. B.$325$
3. C.$346$
4. D.$296$

Let $\mathrm{A}(0,1),\mathrm{B}(1,1)$ and $\mathrm{C}(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $\mathrm{D}$. If the focus of the parabola $y^2=4ax$ passing through $\mathrm{D}$ is $(\alpha +\beta \sqrt{2},0)$, where $\alpha$ and $\beta$ are rational numbers, then $\frac{\alpha }{{\beta }^2}$ is equal to :

1. A.$\frac{9}{2}$
2. B.$12$
3. C.$6$
4. D.$8$

The ordinates of the points $\mathrm{P}$ and $\mathrm{Q}$ on the parabola with focus $(3,0)$ and directrix $x=-3$ are in the ratio $3:1$. If $\mathrm{R}(\alpha ,\beta )$ is the point of intersection of the tangents to the parabola at $\mathrm{P}$ and $\mathrm{Q}$, then $\frac{{\beta }^2}{\alpha }$ is equal to ___________.

Let the tangent to the curve $x^2+2x-4y+9=0$ at the point $\text{P}(1,3)$ on it meet the $y$-axis at A. Let the line passing through P and parallel to the line $x-3y=6$ meet the parabola $y^2=4x$ at B. If B lies on the line $2x-3y=8$, then $(\text{AB}{)}^2$ is equal to ________.

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$, then the eccentricity of the ellipse $25\left({\beta }^2{x}^2+{\alpha }^2{y}^2\right)={\alpha }^2{\beta }^2$ is :

1. A.$\frac{\sqrt{129}}{12}$
2. B.$\frac{3\sqrt{15}}{12}$
3. C.$\frac{\sqrt{119}}{12}$
4. D.$\frac{\sqrt{117}}{12}$

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2x^2-2y^2=1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is ________.