UFM Pure

jee-main 2021 Q80 Circle-Conic Interaction with Tangency or Intersection View

The locus of the midpoints of the chord of the circle, $x ^ { 2 } + y ^ { 2 } = 25$ which is tangent to the hyperbola, $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ is:

jee-main 2021 Q83 Locus and Trajectory Derivation View

The locus of the point of intersection of the lines $( \sqrt { 3 } ) k x + k y - 4 \sqrt { 3 } = 0$ and $\sqrt { 3 } x - y - 4 ( \sqrt { 3 } ) k = 0$ is a conic, whose eccentricity is

jee-main 2021 Q86 Tangent and Normal Line Problems View

Let $P ( a \sec \theta , b \tan \theta )$ and $Q ( a \sec \phi , b \tan \phi )$ where $\theta + \phi = \frac { \pi } { 2 }$, be two points on the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If the ordinate of the point of intersection of normals at $P$ and $Q$ is $- k \left( \frac { a ^ { 2 } + b ^ { 2 } } { 2 b } \right)$, then $k$ is equal to

jee-main 2022 Q63 Tangent and Normal Line Problems View

If $m$ is the slope of a common tangent to the curves $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 12$, then $12 \mathrm {~m} ^ { 2 }$ is equal to
(1) 6
(2) 9
(3) 10
(4) 12

jee-main 2022 Q64 Locus and Trajectory Derivation View

The locus of the mid-point of the line segment joining the point $( 4,3 )$ and the points on the ellipse $x ^ { 2 } + 2 y ^ { 2 } = 4$ is an ellipse with eccentricity
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 1 } { 2 \sqrt { 2 } }$
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $\frac { 1 } { 2 }$

jee-main 2022 Q64 Eccentricity or Asymptote Computation View

Let a line $L$ pass through the point of intersection of the lines $b x + 10 y - 8 = 0$ and $2 x - 3 y = 0$, $b \in R - \left\{ \frac { 4 } { 3 } \right\}$. If the line $L$ also passes through the point $( 1,1 )$ and touches the circle $17 \left( x ^ { 2 } + y ^ { 2 } \right) = 16$, then the eccentricity of the ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ is
(1) $\frac { 2 } { \sqrt { 5 } }$
(2) $\sqrt { \frac { 3 } { 5 } }$
(3) $\frac { 1 } { \sqrt { 5 } }$
(4) $\sqrt { \frac { 2 } { 5 } }$

jee-main 2022 Q64 Confocal or Related Conic Construction View

Let the hyperbola $H : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ pass through the point $( 2 \sqrt { 2 } , - 2 \sqrt { 2 } )$. A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is $e$ times the length of the latus rectum of $H$, where $e$ is the eccentricity of $H$, then which of the following points lies on the parabola?
(1) $( 2 \sqrt { 3 } , 3 \sqrt { 2 } )$
(2) $( 3 \sqrt { 3 } , - 6 \sqrt { 2 } )$
(3) $( \sqrt { 3 } , - \sqrt { 6 } )$
(4) $( 3 \sqrt { 6 } , 6 \sqrt { 2 } )$

jee-main 2022 Q65 Tangent and Normal Line Problems View

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) $( 15 , - 2 \sqrt { 3 } )$
(2) $( 9,2 \sqrt { 3 } )$
(3) $( - 1,9 \sqrt { 3 } )$
(4) $( - 1,6 \sqrt { 3 } )$

jee-main 2022 Q65 Tangent and Normal Line Problems View

Let the eccentricity of the hyperbola $H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6\sqrt{2}$. If $y = 2x + c$ is a tangent to the hyperbola $H$, then the value of $c^2$ is equal to
(1) 18
(2) 20
(3) 24
(4) 32

jee-main 2022 Q65 Triangle or Quadrilateral Area and Perimeter with Foci View

Let the focal chord of the parabola $P : y ^ { 2 } = 4 x$ along the line $L : y = m x + c , m > 0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the hyperbola $H : x ^ { 2 } - y ^ { 2 } = 4$. If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive $x$-axis, then the area of the quadrilateral $O M F N$ is
(1) $2 \sqrt { 6 }$
(2) $2 \sqrt { 14 }$
(3) $4 \sqrt { 6 }$
(4) $4 \sqrt { 14 }$

jee-main 2022 Q65 Tangent and Normal Line Problems View

Let the tangent drawn to the parabola $y ^ { 2 } = 24 x$ at the point $( \alpha , \beta )$ is perpendicular to the line $2 x + 2 y = 5$. Then the normal to the hyperbola $\frac { x ^ { 2 } } { \alpha ^ { 2 } } - \frac { y ^ { 2 } } { \beta ^ { 2 } } = 1$ at the point $( \alpha + 4 , \beta + 4 )$ does NOT pass through the point:
(1) $( 25,10 )$
(2) $( 20,12 )$
(3) $( 30,8 )$
(4) $( 15,13 )$

jee-main 2022 Q66 Eccentricity or Asymptote Computation View

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) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } } { 4 }$

jee-main 2022 Q66 Focal Chord and Parabola Segment Relations View

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 x$ such that it subtends an angle of $\frac { \pi } { 2 }$ at the point $( 3,0 )$. Let the line segment $P Q$ be also a focal chord of the ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , a ^ { 2 } > b ^ { 2 }$. If $e$ is the eccentricity of the ellipse $E$, then the value of $\frac { 1 } { e ^ { 2 } }$ is equal to
(1) $1 + \sqrt { 2 }$
(2) $3 + 2 \sqrt { 2 }$
(3) $1 + 2 \sqrt { 3 }$
(4) $4 + 5 \sqrt { 3 }$

jee-main 2022 Q66 Tangent and Normal Line Problems View

The acute angle between the pair of tangents drawn to the ellipse $2 x ^ { 2 } + 3 y ^ { 2 } = 5$ from the point $(1, 3)$ is
(1) $\tan ^ { - 1 } \frac { 16 } { 7 \sqrt { 5 } }$
(2) $\tan ^ { - 1 } \frac { 24 } { 7 \sqrt { 5 } }$
(3) $\tan ^ { - 1 } \frac { 32 } { 7 \sqrt { 5 } }$
(4) $\tan ^ { - 1 } \frac { 3 + 8 \sqrt { 5 } } { 35 }$

jee-main 2022 Q67 Tangent and Normal Line Problems View

Let $\lambda x - 2 y = \mu$ be a tangent to the hyperbola $a ^ { 2 } x ^ { 2 } - y ^ { 2 } = b ^ { 2 }$. Then $\left( \frac { \lambda } { a } \right) ^ { 2 } - \left( \frac { \mu } { b } \right) ^ { 2 }$ is equal to
(1) - 2
(2) - 4
(3) 2
(4) 4

jee-main 2022 Q67 Eccentricity or Asymptote Computation View

If the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ meets the line $\frac{x}{7} + \frac{y}{2\sqrt{6}} = 1$ on the $x$-axis and the line $\frac{x}{7} - \frac{y}{2\sqrt{6}} = 1$ on the $y$-axis, then the eccentricity of the ellipse is
(1) $\frac{5}{7}$
(2) $\frac{2\sqrt{6}}{7}$
(3) $\frac{3}{7}$
(4) $\frac{2\sqrt{5}}{7}$

jee-main 2022 Q67 Equation Determination from Geometric Conditions View

If the line $x - 1 = 0$, is a directrix of the hyperbola $k x ^ { 2 } - y ^ { 2 } = 6$, then the hyperbola passes through the point
(1) $\left( - 2 \sqrt { 5 } , 6 \right)$
(2) $\left( - \sqrt { 5 } , 3 \right)$
(3) $\left( \sqrt { 5 } , - 2 \right)$
(4) $\left( 2 \sqrt { 5 } , 3 \sqrt { 6 } \right)$

jee-main 2022 Q67 Triangle or Quadrilateral Area and Perimeter with Foci View

If the tangents drawn at the points $P$ and $Q$ on the parabola $y^2 = 2x - 3$ intersect at the point $R(0, 1)$, then the orthocentre of the triangle $PQR$ is
(1) $(0, 1)$
(2) $(2, -1)$
(3) $(6, 3)$
(4) $(2, 1)$

jee-main 2022 Q67 Equation Determination from Geometric Conditions View

If the equation of the parabola, whose vertex is at $( 5,4 )$ and the directrix is $3 x + y - 29 = 0$, is $x ^ { 2 } + a y ^ { 2 } + b x y + c x + d y + k = 0$, then $a + b + c + d + k$ is equal to
(1) 575
(2) - 575
(3) 576
(4) - 576

jee-main 2022 Q68 Eccentricity or Asymptote Computation View

Let the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{7} = 1$ and the hyperbola $\frac{x^2}{144} - \frac{y^2}{\alpha} = \frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is:
(1) $\frac{32}{9}$
(2) $\frac{18}{5}$
(3) $\frac{27}{4}$
(4) $\frac{27}{10}$

jee-main 2022 Q83 Tangent and Normal Line Problems View

A common tangent $T$ to the curves $C _ { 1 } : \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1$ and $C _ { 2 } : \frac { x ^ { 2 } } { 42 } - \frac { y ^ { 2 } } { 143 } = 1$ does not pass through the fourth quadrant. If $T$ touches $C _ { 1 }$ at $\left( x _ { 1 } , y _ { 1 } \right)$ and $C _ { 2 }$ at $\left( x _ { 2 } , y _ { 2 } \right)$, then $\left| 2 x _ { 1 } + x _ { 2 } \right|$ is equal to $\_\_\_\_$ .

jee-main 2022 Q84 Equation Determination from Geometric Conditions View

An ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the vertices of the hyperbola $H : \frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 64 } = - 1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac { 1 } { 2 }$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113 l$ is equal to $\_\_\_\_$ .

jee-main 2022 Q85 Tangent and Normal Line Problems View

Let the common tangents to the curves $4 \left( x ^ { 2 } + y ^ { 2 } \right) = 9$ and $y ^ { 2 } = 4 x$ 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 $\_\_\_\_$.

jee-main 2022 Q86 Eccentricity or Asymptote Computation View

Let the hyperbola $H : \frac { x ^ { 2 } } { a ^ { 2 } } - y ^ { 2 } = 1$ and the ellipse $E : 3 x ^ { 2 } + 4 y ^ { 2 } = 12$ 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 $\_\_\_\_$.

jee-main 2023 Q64 Locus and Trajectory Derivation View

Let a tangent to the curve $y ^ { 2 } = 24 x$ meet the curve $x y = 2$ at the points $A$ and $B$. Then the midpoints of such line segments $A B$ lie on a parabola with the
(1) directrix $4 x = 3$
(2) directrix $4 x = - 3$
(3) Length of latus rectum $\frac { 3 } { 2 }$
(4) Length of latus rectum 2

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