UFM Pure

jee-main 2020 Q57 Eccentricity or Asymptote Computation View

If $e _ { 1 }$ and $e _ { 2 }$ are the eccentricities of the ellipse $\frac { x ^ { 2 } } { 18 } + \frac { y ^ { 2 } } { 4 } = 1$ and the hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ respectively and $\left( e _ { 1 } , e _ { 2 } \right)$ is a point on the ellipse $15 x ^ { 2 } + 3 y ^ { 2 } = k$, then the value of $k$ is equal to
(1) 16
(2) 17
(3) 15
(4) 14

jee-main 2020 Q57 Equation Determination from Geometric Conditions View

A hyperbola having the transverse axis of length, $\sqrt { 2 }$ has the same foci as that of the ellipse, $3 x ^ { 2 } + 4 y ^ { 2 } = 12$ then this hyperbola does not pass through which of the following points?
(1) $\left( \frac { 1 } { \sqrt { 2 } } , 0 \right)$
(2) $\left( - \sqrt { \frac { 3 } { 2 } } , 1 \right)$
(3) $\left( 1 , - \frac { 1 } { \sqrt { 2 } } \right)$
(4) $\left( \sqrt { \frac { 3 } { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$

jee-main 2020 Q57 Focal Distance and Point-on-Conic Metric Computation View

Let $e _ { 1 }$ and $e _ { 2 }$ be the eccentricities of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( b < 5 )$ and the hyperbola $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ respectively satisfying $\mathrm { e } _ { 1 } \mathrm { e } _ { 2 } = 1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $( \alpha , \beta )$ is equal to:
(1) $( 8,10 )$
(2) $\left( \frac { 20 } { 3 } , 12 \right)$
(3) $( 8,12 )$
(4) $\left( \frac { 24 } { 5 } , 10 \right)$

jee-main 2020 Q57 Equation Determination from Geometric Conditions View

Let $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b )$ be a given ellipse, length of whose latus rectum is 10 . If its eccentricity is the maximum value of the function, $\phi ( t ) = \frac { 5 } { 12 } + t - t ^ { 2 }$, then $a ^ { 2 } + b ^ { 2 }$ is equal to:
(1) 145
(2) 116
(3) 126
(4) 135

jee-main 2020 Q57 Tangent and Normal Line Problems View

Let $x = 4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac { 1 } { 2 }$. If $P ( 1 , \beta ) , \beta > 0$ is a point on this ellipse, then the equation of the normal to it at $P$ is
(1) $4 x - 3 y = 2$
(2) $8 x - 2 y = 5$
(3) $7 x - 4 y = 1$
(4) $4 x - 2 y = 1$

jee-main 2020 Q57 Optimization on Conics View

If the point $P$ on the curve, $4 x ^ { 2 } + 5 y ^ { 2 } = 20$ is farthest from the point $Q ( 0 , - 4 )$, then $PQ ^ { 2 }$ is equal to
(1) 36
(2) 48
(3) 21
(4) 29

jee-main 2020 Q57 Tangent and Normal Line Problems View

Let $L _ { 1 }$ be a tangent to the parabola $y ^ { 2 } = 4 ( x + 1 )$ and $L _ { 2 }$ be a tangent to the parabola $y ^ { 2 } = 8 ( x + 2 )$ such that $L _ { 1 }$ and $L _ { 2 }$ intersect at right angles. Then $L _ { 1 }$ and $L _ { 2 }$ meet on the straight line:
(1) $x + 3 = 0$
(2) $2 x + 1 = 0$
(3) $x + 2 = 0$
(4) $x + 2 y = 0$

jee-main 2020 Q57 Eccentricity or Asymptote Computation View

If the normal at an end of latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies:
(1) $\mathrm{e}^{4}+2\mathrm{e}^{2}-1=0$
(2) $\mathrm{e}^{2}+\mathrm{e}-1=0$
(3) $\mathrm{e}^{4}+\mathrm{e}^{2}-1=0$
(4) $\mathrm{e}^{2}+2\mathrm{e}-1=0$

jee-main 2020 Q58 Eccentricity or Asymptote Computation View

The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac { 4 } { \sqrt { 3 } }$. If this ellipse touches the line $x + 6 y = 8$ then its eccentricity is:
(1) $\frac { 1 } { 2 } \sqrt { \frac { 11 } { 3 } }$
(2) $\sqrt { \frac { 5 } { 6 } }$
(3) $\frac { 1 } { 2 } \sqrt { \frac { 5 } { 3 } }$
(4) $\frac { 1 } { 3 } \sqrt { \frac { 11 } { 3 } }$

jee-main 2020 Q58 Eccentricity or Asymptote Computation View

Let $P ( 3,3 )$ be a point on the hyperbola, $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If the normal to it at $P$ intersects the $x$-axis at $( 9,0 )$ and $e$ is its eccentricity, then the ordered pair $\left( a ^ { 2 } , e ^ { 2 } \right)$ is equal to:
(1) $\left( \frac { 9 } { 2 } , 3 \right)$
(2) $\left( \frac { 3 } { 2 } , 2 \right)$
(3) $\left( \frac { 9 } { 2 } , 2 \right)$
(4) $( 9,3 )$

jee-main 2020 Q58 Circle-Conic Interaction with Tangency or Intersection View

If the line $y = mx + c$ is a common tangent to the hyperbola $\frac{x^2}{100} - \frac{y^2}{64} = 1$ and the circle $x^2 + y^2 = 36$, then which one of the following is true?
(1) $c^2 = 369$
(2) $5m = 4$
(3) $4c^2 = 369$
(4) $8m + 5 = 0$

jee-main 2020 Q58 Locus and Trajectory Derivation View

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } = 1$ from any of its foci?
(1) $( - 2 , \sqrt { 3 } )$
(2) $( - 1 , \sqrt { 2 } )$
(3) $( - 1 , \sqrt { 3 } )$
(4) $( 1,2 )$

jee-main 2020 Q59 Equation Determination from Geometric Conditions View

If $3 x + 4 y = 12 \sqrt { 2 }$ is a tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 9 } = 1$ for some $a \in R$, then the distance between the foci of the ellipse is
(1) $2 \sqrt { 7 }$
(2) 4
(3) $2 \sqrt { 5 }$
(4) $2 \sqrt { 2 }$

jee-main 2020 Q59 Eccentricity or Asymptote Computation View

For some $\theta \in \left( 0 , \frac { \pi } { 2 } \right)$, if the eccentricity of the hyperbola, $x ^ { 2 } - y ^ { 2 } \sec ^ { 2 } \theta = 10$ is $\sqrt { 5 }$ times the eccentricity of the ellipse, $x ^ { 2 } \sec ^ { 2 } \theta + y ^ { 2 } = 5$, then the length of the latus rectum of the ellipse, is
(1) $2 \sqrt { 6 }$
(2) $\sqrt { 30 }$
(3) $\frac { 2 \sqrt { 5 } } { 3 }$
(4) $\frac { 4 \sqrt { 5 } } { 3 }$

jee-main 2021 Q65 Locus and Trajectory Derivation View

Let $C$ be the locus of the mirror image of a point on the parabola $y ^ { 2 } = 4x$ with respect to the line $y = x$. Then the equation of tangent to $C$ at $P ( 2,1 )$ is :
(1) $x - y = 1$
(2) $2x + y = 5$
(3) $x + 3y = 5$
(4) $x + 2y = 4$

jee-main 2021 Q65 Triangle or Quadrilateral Area and Perimeter with Foci View

Let the tangent to the parabola $S : y ^ { 2 } = 2 x$ at the point $P ( 2,2 )$ meet the $x$-axis at $Q$ and normal at it meet the parabola $S$ at the point $R$. Then the area (in sq. units) of the triangle $P Q R$ is equal to:
(1) $\frac { 25 } { 2 }$
(2) $\frac { 35 } { 2 }$
(3) $\frac { 15 } { 2 }$
(4) 25

jee-main 2021 Q65 Eccentricity or Asymptote Computation View

Let $E _ { 1 } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , a > b$. Let $E _ { 2 }$ be another ellipse such that it touches the end points of major axis of $E _ { 1 }$ and the foci of $E _ { 2 }$ are the end points of minor axis of $E _ { 1 }$. If $E _ { 1 }$ and $E _ { 2 }$ have same eccentricities, then its value is:
(1) $\frac { - 1 + \sqrt { 5 } } { 2 }$
(2) $\frac { - 1 + \sqrt { 8 } } { 2 }$
(3) $\frac { - 1 + \sqrt { 3 } } { 2 }$
(4) $\frac { - 1 + \sqrt { 6 } } { 2 }$

jee-main 2021 Q66 Locus and Trajectory Derivation View

The locus of the mid-point of the line segment joining the focus of the parabola $y ^ { 2 } = 4 a x$ to a moving point of the parabola, is another parabola whose directrix is:
(1) $x = a$
(2) $x = 0$
(3) $x = - \frac { a } { 2 }$
(4) $x = \frac { a } { 2 }$

jee-main 2021 Q66 Circle-Conic Interaction with Tangency or Intersection View

If the points of intersection of the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the circle $x ^ { 2 } + y ^ { 2 } = 4b , b > 4$ lie on the curve $y ^ { 2 } = 3x ^ { 2 }$, then $b$ is equal to :
(1) 12
(2) 5
(3) 6
(4) 10

jee-main 2021 Q66 Optimization on Conics View

Let a tangent be drawn to the ellipse $\frac { x ^ { 2 } } { 27 } + y ^ { 2 } = 1$ at $( 3 \sqrt { 3 } \cos \theta , \sin \theta )$ where $\theta \in \left( 0 , \frac { \pi } { 2 } \right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to:
(1) $\frac { \pi } { 8 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 6 }$
(4) $\frac { \pi } { 3 }$

jee-main 2021 Q66 Tangent and Normal Line Problems View

Let a line $L : 2 x + y = k , k > 0$ be a tangent to the hyperbola $x ^ { 2 } - y ^ { 2 } = 3$. If $L$ is also a tangent to the parabola $y ^ { 2 } = \alpha x$, then $\alpha$ is equal to:
(1) 12
(2) - 12
(3) 24
(4) - 24

jee-main 2021 Q66 Chord Properties and Midpoint Problems View

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 $(PQ) ^ { 2 }$ is equal to :
(1) $\frac { 25 } { 2 }$
(2) $\frac { 75 } { 8 }$
(3) $\frac { 125 } { 16 }$
(4) $\frac { 15 } { 2 }$

jee-main 2021 Q66 Equation Determination from Geometric Conditions View

A hyperbola passes through the foci of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:
(1) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$
(2) $x ^ { 2 } - y ^ { 2 } = 9$
(3) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$
(4) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$

jee-main 2021 Q67 Triangle or Quadrilateral Area and Perimeter with Foci View

Consider a hyperbola $H : x ^ { 2 } - 2 y ^ { 2 } = 4$. Let the tangent at a point $P ( 4 , \sqrt { 6 } )$ meet the $x$-axis at $Q$ and latus rectum at $R \left( x _ { 1 } , y _ { 1 } \right) , x _ { 1 } > 0$. If $F$ is a focus of $H$ which is nearer to the point $P$, then the area of $\triangle QFR$ (in sq. units) is equal to
(1) $4 \sqrt { 6 }$
(2) $\sqrt { 6 } - 1$
(3) $\frac { 7 } { \sqrt { 6 } } - 2$
(4) $4 \sqrt { 6 } - 1$

jee-main 2021 Q69 Circle-Conic Interaction with Tangency or Intersection View

If a tangent to the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ meets the tangents at the extremities of its major axis at $B$ and $C$, then the circle with $B C$ as diameter passes through the point.
(1) $( \sqrt { 3 } , 0 )$
(2) $( \sqrt { 2 } , 0 )$
(3) $( 1,1 )$
(4) $( - 1,1 )$

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