Conic Sections · JEE Main previous year questions

Let the eccentricity of the ellipse $x^2+a^2{y}^2=25a^2$ be b times the eccentricity of the hyperbola $x^2-a^2{y}^2=5$, where a is the minimum distance between the curves $y={\mathrm{e}}^x$ and $y={\log }_{\mathrm{e}}x$. Then $a^2+\frac{1}{b^2}$ is equal to :

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

Let PQ be a focal chord of length 6.25 units of the parabola $y^2=4x$. If O is the vertex of the parabola, then 10 times the area (in sq. units) of $\Delta \mathrm{POQ}$ is equal to ____ .

Let $a>0,b>0$. Let $e$ and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Let $e^{\prime }$ and $l^{\prime }$ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If $e^2=\frac{11}{14}l$ and $(e^{\prime }{)}^2=\frac{11}{8}{l}^{\prime }$, then the value of $77a+44b$ is equal to :

1. A.$100$
2. B.$110$
3. C.$120$
4. D.$130$

If vertex of a parabola is $(2,-1)$ and the equation of its directrix is $4x-3y=21$, then the length of its latus rectum is :

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

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$, be $\frac{1}{4}$. If this ellipse passes through the point $\left(-4\sqrt{\frac{2}{5}},3\right)$, then $a^2+b^2$ is equal to :

1. A.29
2. B.31
3. C.32
4. D.34

If the equation of the parabola, whose vertex is at $(5,4)$ and the directrix is $3x+y-29=0$, is $x^2+a{y}^2+bxy+cx+dy+k=0$, then $a+b+c+d+k$ is equal to :

1. A.$575$
2. B.$-575$
3. C.$576$
4. D.$-576$

Let the normal at the point $P$ on the parabola $y^2=6x$ pass through the point $(5,-8)$. If the tangent at $P$ to the parabola intersects its directrix at the point $Q$, then the ordinate of the point $Q$ is :

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

Let the common tangents to the curves $4(x^2+y^2)=9$ and $y^2=4x$ intersect at the point $Q$. Let an ellipse, centered at the origin $O$, has lengths of semi-minor and semi-major axes equal to $OQ$ and 6, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\frac{l}{e^2}$ is equal to ____.

The locus of the mid point of the line segment joining the point $(4,3)$ and the points on the ellipse $x^2+2y^2=4$ is an ellipse with eccentricity :

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

The normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{9}=1$ at the point $(8,3\sqrt{3})$ on it passes through the point :

1. A.$(15,-2\sqrt{3})$
2. B.$(9,2\sqrt{3})$
3. C.$(-1,9\sqrt{3})$
4. D.$(-1,6\sqrt{3})$

Let a line ${\mathrm{L}}_1$ be tangent to the hyperbola $\frac{x^2}{16}-\frac{y^2}{4}=1$ and let ${\mathrm{L}}_2$ be the line passing through the origin and perpendicular to ${\mathrm{L}}_1$. If the locus of the point of intersection of ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ is $(x^2+y^2{)}^2=\alpha x^2+\beta y^2$, then $\alpha +\beta$ is equal to ________.

The area of the region enclosed between the parabolas $y^2=2x-1$ and $y^2=4x-3$ is

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

If the line $y=4+kx$, $k>0$, is the tangent to the parabola $y=x-x^2$ at the point $\mathrm{P}$ and $\mathrm{V}$ is the vertex of the parabola, then the slope of the line through $\mathrm{P}$ and $\mathrm{V}$ is :

1. A.$\frac{3}{2}$
2. B.$\frac{26}{9}$
3. C.$\frac{5}{2}$
4. D.$\frac{23}{6}$

Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be $\frac{5}{4}$. If the equation of the normal at the point $\left(\frac{8}{\sqrt{5}},\frac{12}{5}\right)$ on the hyperbola is $8\sqrt{5}x+\beta y=\lambda$, then $\lambda -\beta$ is equal to ____.

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :

1. A.length of latus rectum 3
2. B.length of latus rectum 6
3. C.focus $\left(\frac{4}{3},0\right)$
4. D.focus $\left(0,\frac{3}{4}\right)$

Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{4}=1$, $a>2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $6\sqrt{3}$. Then the eccentricity of the ellipse is :

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

The area (in sq. units) of the region enclosed between the parabola $y^2=2x$ and the line $x+y=4$ is ____.

Let the hyperbola $H:\frac{x^2}{a^2}-y^2=1$ and the ellipse $E:3x^2+4y^2=12$ be such that the length of latus rectum of H is equal to the length of latus rectum of E. If $e_H$ and $e_E$ are the eccentricities of H and E respectively, then the value of $12\left(e_H^2+e_E^2\right)$ is equal to ____.

Let ${\mathrm{P}}_1$ be a parabola with vertex $(3,2)$ and focus $(4,4)$ and ${\mathrm{P}}_2$ be its mirror image with respect to the line $x+2y=6$. Then the directrix of ${\mathrm{P}}_2$ is $x+2y=$ ____.

Consider the parabola with vertex $\left(\frac{1}{2},\frac{3}{4}\right)$ and the directrix $y=\frac{1}{2}$. Let P be the point where the parabola meets the line $x=-\frac{1}{2}$. If the normal to the parabola at P intersects the parabola again at the point Q, then $(\mathrm{PQ}{)}^2$ is equal to :

1. A.$\frac{75}{8}$
2. B.$\frac{125}{16}$
3. C.$\frac{25}{2}$
4. D.$\frac{15}{2}$

The locus of the mid points of the chords of the hyperbola $x^2-y^2=4$, which touch the parabola $y^2=8x$, is :

1. A.$y^3(x-2)=x^2$
2. B.$x^3(x-2)=y^2$
3. C.$y^2(x-2)=x^3$
4. D.$x^2(x-2)=y^3$

The point $P(-2\sqrt{6},\sqrt{3})$ lies on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ having eccentricity $\frac{\sqrt{5}}{2}$. If the tangent and normal at P to the hyperbola intersect its conjugate axis at the point Q and R respectively, then QR is equal to :

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

Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3, $-4$), one focus at (4, $-4$) and one vertex at (5, $-4$). If $mx-y=4$, m > 0 is a tangent to the ellipse E, then the value of $5m^2$ is equal to ____.

The locus of the centroid of the triangle formed by any point P on the hyperbola $16x^2-9y^2+32x+36y-164=0$, and its foci is :

1. A.$16x^2-9y^2+32x+36y-36=0$
2. B.$9x^2-16y^2+36x+32y-144=0$
3. C.$16x^2-9y^2+32x+36y-144=0$
4. D.$9x^2-16y^2+36x+32y-36=0$

Let a parabola P be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from O(0, 0) to the parabola P which meet P at S and R, then the area (in sq. units) of $\Delta$SOR is equal to :

1. A.$16\sqrt{2}$
2. B.16
3. C.32
4. D.$8\sqrt{2}$