UFM Pure

jee-main 2023 Q65 Eccentricity or Asymptote Computation View

If the maximum distance of normal to the ellipse $\frac{x^2}{4} + \frac{y^2}{b^2} = 1$, $b < 2$, from the origin is 1, then the eccentricity of the ellipse is:
(1) $\frac{1}{\sqrt{2}}$
(2) $\frac{\sqrt{3}}{2}$
(3) $\frac{1}{2}$
(4) $\frac{\sqrt{3}}{4}$

jee-main 2023 Q68 Circle-Conic Interaction with Tangency or Intersection View

Let a circle of radius 4 be concentric to the ellipse $15 x ^ { 2 } + 19 y ^ { 2 } = 285$. Then the common tangents are inclined to the minor axis of the ellipse at the angle
(1) $\frac { \pi } { 3 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 6 }$
(4) $\frac { \pi } { 12 }$

jee-main 2023 Q70 Eccentricity or Asymptote Computation View

Let H be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is:
(1) 3
(2) $\frac{5}{2}$
(3) 2
(4) $\frac{3}{2}$

jee-main 2023 Q70 Tangent and Normal Line Problems View

The vertices of a hyperbola H are $( \pm 6,0 )$ and its eccentricity is $\frac { \sqrt { 5 } } { 2 }$. Let N be the normal to H at a point in the first quadrant and parallel to the line $\sqrt { 2 } x + y = 2 \sqrt { 2 }$. If $d$ is the length of the line segment of N between H and the $y$-axis then $d ^ { 2 }$ is equal to $\_\_\_\_$ .

jee-main 2023 Q71 Eccentricity or Asymptote Computation View

Let the eccentricity of an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is reciprocal to that of the hyperbola $2x^{2} - 2y^{2} = 1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is $\_\_\_\_$.

jee-main 2023 Q71 Focal Chord and Parabola Segment Relations View

Let $R$ be the focus of the parabola $y ^ { 2 } = 20 x$ and the line $y = m x + c$ intersect the parabola at two points $P$ and $Q$. Let the point $G ( 10 , 10 )$ be the centroid of the triangle $P Q R$. If $c - m = 6$, then $P Q ^ { 2 }$ is
(1) 296
(2) 325
(3) 317
(4) 346

jee-main 2023 Q71 Chord Properties and Midpoint Problems View

Let $P \left( \frac { 2 \sqrt { 3 } } { \sqrt { 7 } } , \frac { 6 } { \sqrt { 7 } } \right) , Q , R$ and $S$ be four points on the ellipse $9 x ^ { 2 } + 4 y ^ { 2 } = 36$. Let $P Q$ and $R S$ be mutually perpendicular and pass through the origin. If $\frac { 1 } { ( P Q ) ^ { 2 } } + \frac { 1 } { ( R S ) ^ { 2 } } = \frac { p } { q }$, where $p$ and $q$ are coprime, then $p + q$ is equal to
(1) 147
(2) 143
(3) 137
(4) 157

jee-main 2023 Q72 Locus and Trajectory Derivation View

The equations of two sides of a variable triangle are $x = 0$ and $y = 3$, and its third side is a tangent to the parabola $y ^ { 2 } = 6 x$. The locus of its circumcentre is:
(1) $4 y ^ { 2 } - 18 y - 3 x - 18 = 0$
(2) $4 y ^ { 2 } + 18 y + 3 x + 18 = 0$
(3) $4 y ^ { 2 } - 18 y + 3 x + 18 = 0$
(4) $4 y ^ { 2 } - 18 y - 3 x + 18 = 0$

jee-main 2024 Q66 Focal Distance and Point-on-Conic Metric Computation View

If the foci of a hyperbola are same as that of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ and the eccentricity of the hyperbola is $\frac { 15 } { 8 }$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt { 2 } , \frac { 14 } { 3 } \sqrt { \frac { 2 } { 5 } }\right)$ on the hyperbola is equal to
(1) $7 \sqrt { \frac { 2 } { 5 } } - \frac { 8 } { 3 }$
(2) $14 \sqrt { \frac { 2 } { 5 } } - \frac { 4 } { 3 }$
(3) $14 \sqrt { \frac { 2 } { 5 } } - \frac { 16 } { 3 }$
(4) $7 \sqrt { \frac { 2 } { 5 } } + \frac { 8 } { 3 }$

jee-main 2024 Q66 Equation Determination from Geometric Conditions View

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) 237
(2) 242
(3) 205
(4) 225

jee-main 2024 Q67 Locus and Trajectory Derivation View

Let $P$ be a point on the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. Let the line passing through $P$ and parallel to $y$-axis meet the circle $x^2 + y^2 = 9$ at point $Q$ such that $P$ and $Q$ 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) $\frac{11}{19}$
(2) $\frac{13}{21}$
(3) $\frac{\sqrt{139}}{23}$
(4) $\frac{\sqrt{13}}{7}$

jee-main 2024 Q67 Eccentricity or Asymptote Computation View

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :
(1) $\frac { \sqrt { 5 } } { 3 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) $\frac { 1 } { \sqrt { 3 } }$
(4) $\frac { 2 } { \sqrt { 5 } }$

jee-main 2024 Q67 Chord Properties and Midpoint Problems View

Let $e _ { 1 }$ be the eccentricity of the hyperbola $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ and $e _ { 2 }$ be the eccentricity of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , a > b$, which passes through the foci of the hyperbola. If $e _ { 1 } e _ { 2 } = 1$, then the length of the chord of the ellipse parallel to the x -axis and passing through $( 0,2 )$ is:
(1) $4 \sqrt { 5 }$
(2) $\frac { 8 \sqrt { 5 } } { 3 }$
(3) $\frac { 10 \sqrt { 5 } } { 3 }$
(4) $3 \sqrt { 5 }$

jee-main 2024 Q67 Triangle or Quadrilateral Area and Perimeter with Foci View

Let $P$ be a point on the hyperbola $H: \frac{x^2}{9} - \frac{y^2}{4} = 1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then, the square of the distance of $P$ from the origin is
(1) 18
(2) 26
(3) 22
(4) 20

jee-main 2024 Q67 Circle-Conic Interaction with Tangency or Intersection View

Consider a hyperbola H having centre at the origin and foci on the x-axis. Let $\mathrm { C } _ { 1 }$ be the circle touching the hyperbola H and having the centre at the origin. Let $\mathrm { C } _ { 2 }$ be the circle touching the hyperbola 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 H is
(1) $\frac { 14 } { 3 }$
(2) $\frac { 28 } { 3 }$
(3) $\frac { 11 } { 3 }$
(4) $\frac { 10 } { 3 }$

jee-main 2024 Q67 Focal Distance and Point-on-Conic Metric Computation View

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) 172
(2) 171
(3) 169
(4) 170

jee-main 2024 Q68 Equation Determination from Geometric Conditions View

Let $P$ be a parabola with vertex $(2, 3)$ and directrix $2x + y = 6$. Let an ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ of eccentricity $\dfrac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then the square of the length of the latus rectum of $E$, is
(1) $\dfrac{385}{8}$
(2) $\dfrac{347}{8}$
(3) $\dfrac{512}{25}$
(4) $\dfrac{656}{25}$

jee-main 2024 Q68 Chord Properties and Midpoint Problems View

The length of the chord of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$, whose mid point is $\left( 1 , \frac { 2 } { 5 } \right)$, is equal to:
(1) $\frac { \sqrt { 1691 } } { 5 }$
(2) $\frac { \sqrt { 2009 } } { 5 }$
(3) $\frac { \sqrt { 1741 } } { 5 }$
(4) $\frac { \sqrt { 1541 } } { 5 }$

jee-main 2024 Q83 Focal Distance and Point-on-Conic Metric Computation View

The length of the latus rectum and directrices of a hyperbola with eccentricity 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 m is the product of the focal distances of the point $\left( x _ { 0 } , y _ { 0 } \right)$, then $4 \mathrm { e } ^ { 2 } + \mathrm { m }$ is equal to $\_\_\_\_$

jee-main 2025 Q3 Eccentricity or Asymptote Computation View

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

jee-main 2025 Q3 Triangle or Quadrilateral Area and Perimeter with Foci View

Let $ABCD$ be a trapezium whose vertices lie on the parabola $y ^ { 2 } = 4 x$. 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) $\frac { 75 } { 4 }$
(2) $\frac { 25 } { 2 }$
(3) $\frac { 125 } { 8 }$
(4) $\frac { 75 } { 8 }$

jee-main 2025 Q9 Triangle or Quadrilateral Area and Perimeter with Foci View

Let $\mathrm { P } ( 4,4 \sqrt { 3 } )$ be a point on the parabola $y ^ { 2 } = 4 \mathrm { a } x$ and PQ be a focal chord of the parabola. If M and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :
(1) $17 \sqrt { 3 }$
(2) $\frac { 263 \sqrt { 3 } } { 8 }$
(3) $\frac { 34 \sqrt { 3 } } { 3 }$
(4) $\frac { 343 \sqrt { 3 } } { 8 }$

jee-main 2025 Q9 Chord Properties and Midpoint Problems View

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) $\frac { 5 } { 3 } \sqrt { 15 }$
(2) $\frac { 1 } { 3 } \sqrt { 15 }$
(3) $\frac { 2 } { 3 } \sqrt { 15 }$
(4) $\sqrt { 15 }$

jee-main 2025 Q10 Triangle or Quadrilateral Area and Perimeter with Foci View

Let the ellipse $\mathrm{E}_1: \frac{x^2}{\mathrm{a}^2} + \frac{y^2}{\mathrm{b}^2} = 1,\ \mathrm{a} > \mathrm{b}$ and $\mathrm{E}_2: \frac{x^2}{\mathrm{A}^2} + \frac{y^2}{\mathrm{B}^2} = 1,\ \mathrm{A} < \mathrm{B}$ have same eccentricity $\frac{1}{\sqrt{3}}$. Let the product of their lengths of latus rectums be $\frac{32}{\sqrt{3}}$, and the distance between the foci of $E_1$ be 4. If $E_1$ and $E_2$ meet at $A, B, C$ and $D$, then the area of the quadrilateral $ABCD$ equals:
(1) $\frac{12\sqrt{6}}{5}$
(2) $6\sqrt{6}$
(3) $\frac{18\sqrt{6}}{5}$
(4) $\frac{24\sqrt{6}}{5}$

jee-main 2025 Q13 Chord Properties and Midpoint Problems View

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) 58
(2) 46
(3) 37
(4) 72

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