UFM Pure

jee-main 2014 Q73 Eccentricity or Asymptote Computation View

If $OB$ is the semi-minor axis of an ellipse, $F _ { 1 }$ and $F _ { 2 }$ are its focii and the angle between $F _ { 1 } B$ and $F _ { 2 } B$ is a right angle, then the square of the eccentricity of the ellipse is
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 2 \sqrt { 2 } }$

jee-main 2016 Q70 Tangent and Normal Line Problems View

The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$. If one of its directrices is $x = -4$, then the equation of the normal to it at $\left(1, \frac{3}{2}\right)$ is:
(1) $4x - 2y = 1$
(2) $4x + 2y = 7$
(3) $x + 2y = 4$
(4) $2y - x = 2$

jee-main 2016 Q71 Eccentricity or Asymptote Computation View

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is: (1) $\frac{4}{3}$ (2) $\frac{4}{\sqrt{3}}$ (3) $\frac{2}{\sqrt{3}}$ (4) $\sqrt{3}$

jee-main 2016 Q73 Locus and Trajectory Derivation View

The centres of those circles which touch the circle, $x^2 + y^2 - 8x - 8y - 4 = 0$, externally and also touch the $x$-axis, lie on:
(1) a circle
(2) an ellipse which is not a circle
(3) a hyperbola
(4) a parabola

jee-main 2016 Q73 Equation Determination from Geometric Conditions View

A hyperbola whose transverse axis is along the major axis of the conic $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 4 } = 4$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $\frac { 3 } { 2 }$, then which of the following points does not lie on the hyperbola?
(1) $( \sqrt { 5 } , 2 \sqrt { 2 } )$
(2) $( 0,2 )$
(3) $( 5,2 \sqrt { 3 } )$
(4) $( \sqrt { 10 } , 2 \sqrt { 3 } )$

jee-main 2016 Q76 Optimization on Conics View

If the tangent at a point on the ellipse $\frac{x^2}{27} + \frac{y^2}{3} = 1$ meets the coordinate axes at $A$ and $B$, and $O$ is the origin, then the minimum area (in sq. units) of the triangle $OAB$ is:
(1) $\frac{9}{2}$
(2) $9$
(3) $9\sqrt{3}$
(4) $\frac{\sqrt{3}}{2}$

jee-main 2017 Q69 Tangent and Normal Line Problems View

The eccentricity of an ellipse whose centre is at the origin is $\dfrac{1}{2}$. If one of its directrices is $x = -4$, then the equation of the normal to it at $\left(1, \dfrac{3}{2}\right)$ is:
(1) $2y - x = 2$
(2) $4x - 2y = 1$
(3) $4x + 2y = 7$
(4) $x + 2y = 4$

jee-main 2017 Q70 Tangent and Normal Line Problems View

A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$. Then the tangent to this hyperbola at $P$ also passes through the point
(1) $(3\sqrt{2}, 2\sqrt{3})$
(2) $(2\sqrt{2}, 3\sqrt{3})$
(3) $(\sqrt{3}, \sqrt{2})$
(4) $(-\sqrt{2}, -\sqrt{3})$

jee-main 2017 Q72 Tangent and Normal Line Problems View

The eccentricity of an ellipse whose centre is at the origin is $\frac { 1 } { 2 }$. If one of its directrices is $x = - 4$, then the equation of the normal to it at $\left( 1 , \frac { 3 } { 2 } \right)$ is:
(1) $4 x - 2 y = 1$
(2) $4 x + 2 y = 7$
(3) $x + 2 y = 4$
(4) $2 y - x = 2$

jee-main 2017 Q73 Tangent and Normal Line Problems View

A hyperbola passes through the point $P ( \sqrt { 2 } , \sqrt { 3 } )$ and has foci at $( \pm 2 , 0 )$. Then the tangent to this hyperbola at $P$ also passes through the point:
(1) $( 3 \sqrt { 2 } , 2 \sqrt { 3 } )$
(2) $( 2 \sqrt { 2 } , 3 \sqrt { 3 } )$
(3) $( \sqrt { 3 } , \sqrt { 2 } )$
(4) $( - \sqrt { 2 } , - \sqrt { 3 } )$

jee-main 2018 Q68 Locus and Trajectory Derivation View

The locus of the point of intersection of the lines $\sqrt { 2 } x - y + 4 \sqrt { 2 } k = 0$ and $\sqrt { 2 } k x + k y - 4 \sqrt { 2 } = 0$ ( $k$ is any non-zero real parameter) is
(1) an ellipse whose eccentricity is $\frac { 1 } { \sqrt { 3 } }$
(2) a hyperbola whose eccentricity is $\sqrt { 3 }$
(3) a hyperbola with length of its transverse axis $8 \sqrt { 2 }$
(4) an ellipse with length of its major axis $8 \sqrt { 2 }$

jee-main 2018 Q70 Optimization on Conics View

Let $P$ be a point on the parabola $x ^ { 2 } = 4 y$. If the distance of $P$ from the center of the circle $x ^ { 2 } + y ^ { 2 } + 6 x + 8 = 0$ is minimum, then the equation of the tangent to the parabola at $P$ is
(1) $x + y + 1 = 0$
(2) $x + 4 y - 2 = 0$
(3) $x + 2 y = 0$
(4) $x - y + 3 = 0$

jee-main 2018 Q71 Eccentricity or Asymptote Computation View

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is $\frac { 3 } { 2 }$ units, then its eccentricity is
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 9 }$
(4) $\frac { 1 } { 3 }$

jee-main 2018 Q72 Triangle or Quadrilateral Area and Perimeter with Foci View

Tangents are drawn to the hyperbola $4 x ^ { 2 } - y ^ { 2 } = 36$ at the points $P$ and $Q$. If these tangents intersect at the point $T ( 0,3 )$ then the area (in sq. units) of $\triangle P T Q$ is:
(1) $36 \sqrt { 5 }$
(2) $45 \sqrt { 5 }$
(3) $54 \sqrt { 3 }$
(4) $60 \sqrt { 3 }$

jee-main 2019 Q70 Focal Chord and Parabola Segment Relations View

If one end of a focal chord of the parabola, $y ^ { 2 } = 16 x$ is at $( 1,4 )$, then the length of this focal chord is
(1) 24
(2) 25
(3) 22
(4) 20

jee-main 2019 Q71 Tangent and Normal Line Problems View

If the line $y = m x + 7 \sqrt { 3 }$ is normal to the hyperbola $\frac { x ^ { 2 } } { 24 } - \frac { y ^ { 2 } } { 18 } = 1$, then a value of $m$ is:
(1) $\frac { \sqrt { 5 } } { 2 }$
(2) $\frac { 3 } { \sqrt { 5 } }$
(3) $\frac { \sqrt { 15 } } { 2 }$
(4) $\frac { 2 } { \sqrt { 5 } }$

jee-main 2019 Q72 Tangent and Normal Line Problems View

The equation of a tangent to the hyperbola, $4 x ^ { 2 } - 5 y ^ { 2 } = 20$, parallel to the line $x - y = 2$, is
(1) $x - y + 7 = 0$
(2) $x - y - 3 = 0$
(3) $x - y + 1 = 0$
(4) $x - y + 9 = 0$

jee-main 2019 Q73 Eccentricity or Asymptote Computation View

A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length 4 along the $x$-axis. Then the eccentricity of the hyperbola is:
(1) $\sqrt{3}$
(2) $\frac{3}{2}$
(3) $\frac{2}{\sqrt{3}}$
(4) 2

jee-main 2020 Q56 Tangent and Normal Line Problems View

If $y = m x + 4$ is a tangent to both the parabolas, $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 2 b y$, then $b$ is equal to
(1) $-32$
(2) $-64$
(3) $-128$
(4) 128

jee-main 2020 Q56 Tangent and Normal Line Problems View

If a hyperbola passes through the point $P(10, 16)$, and it has vertices at $(\pm 6, 0)$, then the equation of the normal to it at $P$ is.
(1) $3x + 4y = 94$
(2) $2x + 5y = 100$
(3) $x + 2y = 42$
(4) $x + 3y = 58$

jee-main 2020 Q56 Tangent and Normal Line Problems View

A line parallel to the straight line $2x - y = 0$ is tangent to the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{2} = 1$ at the point $(x_{1}, y_{1})$. Then $x_{1}^{2} + 5y_{1}^{2}$ is equal to
(1) 6
(2) 8
(3) 10
(4) 5

jee-main 2020 Q56 Chord Properties and Midpoint Problems View

Let P be a point on the parabola, $y ^ { 2 } = 12 x$ and N be the foot of the perpendicular drawn from $P$, on the axis of the parabola. A line is now drawn through the mid-point $M$ of $P N$, parallel to its axis which meets the parabola at $Q$. If the $y$-intercept of the line NQ is $\frac { 4 } { 3 }$, then:
(1) $P N = 4$
(2) $M Q = \frac { 1 } { 3 }$
(3) $M Q = \frac { 1 } { 4 }$
(4) $P N = 3$

jee-main 2020 Q56 Focal Distance and Point-on-Conic Metric Computation View

If the co-ordinates of two points $A$ and $B$ are $( \sqrt { 7 } , 0 )$ and $( - \sqrt { 7 } , 0 )$ respectively and $P$ is any point on the conic, $9 x ^ { 2 } + 16 y ^ { 2 } = 144$, then $PA + PB$ is equal to :
(1) 16
(2) 8
(3) 6
(4) 9

jee-main 2020 Q57 Eccentricity or Asymptote Computation View

If the distance between the foci of an ellipse is 6 and the distance between its directrix is 12, then the length of its latus rectum is
(1) $\sqrt { 3 }$
(2) $3 \sqrt { 2 }$
(3) $\frac { 3 } { \sqrt { 2 } }$
(4) $2 \sqrt { 3 }$

jee-main 2020 Q57 Focal Chord and Parabola Segment Relations View

If one end of a focal chord $AB$ of the parabola $y ^ { 2 } = 8 x$ is at $A \left( \frac { 1 } { 2 } , - 2 \right)$, then the equation of the tangent to it at $B$ is:
(1) $2 x + y - 24 = 0$
(2) $x - 2 y + 8 = 0$
(3) $x + 2 y + 8 = 0$
(4) $2 x - y - 24 = 0$

UFM Pure

Sequences and series, recurrence and convergence 375

Roots of polynomials 146

Polar coordinates 22

Conic sections 291

Taylor series 183

Hyperbolic functions 10

Integration using inverse trig and hyperbolic functions 6

Integration with Partial Fractions 7

Vectors: Lines & Planes 293

Vectors: Cross Product & Distances 20

First order differential equations (integrating factor) 60

Complex numbers 2 113

Second order differential equations 127

Systems of differential equations 41