Conic Sections · JEE Main previous year questions

If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, whose mid point is $\left(\frac{5}{2},\frac{1}{2}\right)$. then $\alpha +\beta$ is equal to :

1. A.$37$
2. B.$46$
3. C.$72$
4. D.$58$

Let $y^2=12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ)=\frac{147}{4}$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2+64y^2-\alpha x-64\sqrt{3}y=\beta$, then $\beta -\alpha$ is equal to ____.

Let ABCD be a trapezium whose vertices lie on the parabola $y^2=4x$. Let the sides AD and BC of the trapezium be parallel to $y$-axis. If the diagonal AC is of length $\frac{25}{4}$ and it passes through the point $(1,0)$, then the area of $ABCD$ is

1. A.$\frac{75}{8}$
2. B.$\frac{125}{8}$
3. C.$\frac{25}{2}$
4. D.$\frac{75}{4}$

Let ${\mathrm{E}}_1:\frac{x^2}{9}+\frac{y^2}{4}=1$ be an ellipse. Ellipses ${\mathrm{E}}_i$ 's are constructed such that their centres and eccentricities are same as that of ${\mathrm{E}}_1$, and the length of minor axis of ${\mathrm{E}}_i$ is the length of major axis of ${\mathrm{E}}_{i+1}(i\ge 1)$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi }\left({\sum }_{i=1}^{\infty }{A}_i\right)$, is equal to ____.

Let the product of the focal distances of the point $\left(\sqrt{3},\frac{1}{2}\right)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is

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

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2},3\right)$ and the directrix $x+2y=0$ is $\alpha x^2+\beta y^2-\gamma xy-30x-60y+225=0$, then $\alpha +\beta +\gamma$ is equal to :

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

The equation of the chord, of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid-point is $(3,1)$ is :

1. A.$5x+16y=31$
2. B.$48x+25y=169$
3. C.$4x+122y=134$
4. D.$25x+101y=176$

Let ${\mathrm{H}}_1:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and ${\mathrm{H}}_2:-\frac{x^2}{{\mathrm{A}}^2}+\frac{y^2}{{\mathrm{B}}^2}=1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1=\sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________ .

The length of the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$, whose mid-point is $\left(1,\frac{1}{2}\right)$, is :

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

Let the foci of a hyperbola be $(1,14)$ and $(1,-12)$. If it passes through the point $(1,6)$, then the length of its latus-rectum is :

1. A.$\frac{25}{6}$
2. B.$\frac{144}{5}$
3. C.$\frac{288}{5}$
4. D.$\frac{24}{5}$

Let $f(x)=x^2+9$, $g(x)=\frac{x}{x-9}$ and $\mathrm{a}=f\circ g(10)$, $\mathrm{b}=g\circ f(3)$. If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{\mathrm{a}}+\frac{y^2}{\mathrm{b}}=1$, then $8e^2+l^2$ is equal to:

1. A.6
2. B.12
3. C.8
4. D.16

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E:\frac{(x-1{)}^2}{100}+\frac{(y-1{)}^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3{\alpha }^2+2{\beta }^2$ is equal to

1. A.$225$
2. B.$237$
3. C.$242$
4. D.$205$

The area (in square units) of the region enclosed by the ellipse $x^2+3y^2=18$ in the first quadrant below the line $y=x$ is

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

Let $A,B$ and $C$ be three points on the parabola $y^2=6x$ and let the line segment $AB$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then ${\left(\frac{AM\cdot BN}{CD}\right)}^2$ is equal to ____.

Let $H:-\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha ,6),\alpha >0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha ,6)$, then ${\alpha }^2+\beta$ is equal to

1. A.$170$
2. B.$171$
3. C.$169$
4. D.$172$

Let $\mathrm{A}$ be the region enclosed by the parabola $y^2=2x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $\mathrm{A}$ is ____.

The length of the latus rectum and directrices of hyperbola with eccentricity $\mathrm{e}$ are 9 and $x=\pm \frac{4}{\sqrt{13}}$, respectively. Let the line $y-\sqrt{3}x+\sqrt{3}=0$ touch this hyperbola at $(x_0,y_0)$. If $\mathrm{m}$ is the product of the focal distances of the point $(x_0,y_0)$, then $4{\mathrm{e}}^2+\mathrm{m}$ is equal to ____.

Let $A$ be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x,y)$ satisfying $A^2+xA+y\mathrm{I}=\mathrm{O}$ lie on a hyperbola, whose transverse axis is parallel to the $x$-axis, eccentricity is $e$ and the length of the latus rectum is $l$, then $e^4+l^4$ is equal to ____.

Let the length of the focal chord $\mathrm{PQ}$ of the parabola $y^2=12x$ be 15 units. If the distance of $\mathrm{PQ}$ from the origin is $\mathrm{p}$, then $10{\mathrm{p}}^2$ is equal to ____.

Consider a hyperbola $\mathrm{H}$ having centre at the origin and foci on the $x$-axis. Let $C_1$ be the circle touching the hyperbola $\mathrm{H}$ and having the centre at the origin. Let $C_2$ be the circle touching the hyperbola $\mathrm{H}$ at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are $36\pi$ and $4\pi$, respectively, then the length (in units) of latus rectum of $\mathrm{H}$ is

1. A.$\frac{28}{3}$
2. B.$\frac{11}{3}$
3. C.$\frac{14}{3}$
4. D.$\frac{10}{3}$

Let $PQ$ be a chord of the parabola $y^2=12x$ and the midpoint of $PQ$ be at $(4,1)$. Then, which of the following point lies on the line passing through the points $\mathrm{P}$ and $\mathrm{Q}$ ?

1. A.$(3,-3)$
2. B.$\left(\frac{1}{2},-20\right)$
3. C.$(2,-9)$
4. D.$\left(\frac{3}{2},-16\right)$

For $0<\theta <\pi /2$, if the eccentricity of the hyperbola $x^2-y^2{\mathrm{cosec}}^2\theta =5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2{\mathrm{cosec}}^2\theta +y^2=5$, then the value of $\theta$ is :

1. A.$\frac{\pi }{6}$
2. B.$\frac{5\pi }{12}$
3. C.$\frac{\pi }{3}$
4. D.$\frac{\pi }{4}$

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\mathrm{a}>\mathrm{b}$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is :

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

Let $\mathrm{P}$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let the line passing through $\mathrm{P}$ and parallel to $y$-axis meet the circle $x^2+y^2=9$ at point $\mathrm{Q}$ such that $\mathrm{P}$ and $\mathrm{Q}$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR:RQ=4:3$ as $P$ moves on the ellipse, is :

1. A.$\frac{13}{21}$
2. B.$\frac{\sqrt{139}}{23}$
3. C.$\frac{\sqrt{13}}{7}$
4. D.$\frac{11}{19}$

Three points $\mathrm{O}(0,0)$, $\mathrm{P}\left(\mathrm{a},{\mathrm{a}}^2\right)$, $\mathrm{Q}\left(-\mathrm{b},{\mathrm{b}}^2\right)$, $\mathrm{a}>0$, $\mathrm{b}>0$, are on the parabola $y=x^2$. Let ${\mathrm{S}}_1$ be the area of the region bounded by the line $\mathrm{PQ}$ and the parabola, and ${\mathrm{S}}_2$ be the area of the triangle $\mathrm{OPQ}$. If the minimum value of $\frac{{\mathrm{S}}_1}{{\mathrm{S}}_2}$ is $\frac{\mathrm{m}}{\mathrm{n}}$, $\gcd (\mathrm{m},\mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to ________ .