Q JEE MAIN 2022

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,...

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Q JEE-Main 2024

Let an ellipse $\frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}

JEE Main Physics Easy

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Q JEE MAIN 2026

Let $\frac{x^2}{f\left(a^2+7 a+3\right)}+\frac{y^2}{f(3 a+15)}=1$ represent an ellipse with major axis along $y$-axis, where $f$ is a strictly decreasing positive function...

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Q JEE MAIN 2026

Let $x=9$ be a directrix of an ellipse $E$, whose centre is at the origin and eccentricity is $\frac{1}{3}$. Let...

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Q JEE MAIN 2026

Let $\mathrm{P}(3 \cos \alpha, 2 \sin \alpha), \alpha \neq 0$, be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1, \mathrm{Q}$ be a...

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Q JEE MAIN 2026

Let a focus of the ellipse $\mathrm{E}: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be $\mathrm{S}(4,0)$ and its eccentricity be $\frac{4}{5}$. If the point $\mathrm{P}(3, \alpha)$...

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Q JEE Main 2026

Let A be the point $(3,0)$ and circles with variable diameter AB touch the circle $x^2+y^2=36$ internally. Let the curve...

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Q Jee Main 2026

Consider the parabola $\mathrm{P}: y^2=4 k x$ and the ellipse $\mathrm{E}: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Let the line segment joining the points of...

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Q Jee Main 2024

Let P be a parabola with vertex $(2,3)$ and directrix $2 x+y=6$. Let an ellipse $E: x_{a_2}{ }^2+y_{b^2}=1, a>b$ of...

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Q JEE MAIN 2024

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...

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Q JEE MAIN 2024

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5 x+7 y=50$. Let the point $P$ divide...

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Q JEE Main 2026

Let $(h, k)$ lie on the circle $\mathrm{C}: x^2+y^2=4$ and the point $(2 h+1,3 k+2)$ lie on an ellipse with...

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Q JEE MAIN 2026

An ellipse has its center at $(1,-2)$, one focus at $(3,-2)$ and one vertex at $(5,-2)$. Then the length of...

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Q JEE MAIN 2026

If the points of intersection of the ellipses $x^2+2 y^2-6 x-12 y+23=0$ and $4 x^2+2 y^2-20 x-12 y+35=0$ lie on...

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Q JEE MAIN 2026

Let each of the two ellipses $\mathrm{E}_1: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ and $\mathrm{E}_2: \frac{x^2}{\mathrm{~A}^2}+\frac{y^2}{\mathrm{~B}^2}=1,(\mathrm{~A}

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Q JEE MAIN_2026

Let the line $y-x=1$ intersect the ellipse $\frac{x^2}{2}+\frac{y^2}{1}=1$ at the points A and B . Then the angle made by...

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Q JEE MAIN 2026

Let $S$ and $S^{\prime}$ be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $P(\alpha, \beta)$ be a point on the ellipse...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

If the line $\alpha x+4 y=\sqrt{7}$, where $\alpha \in \mathbf{R}$, touches the ellipse $3 x^2+4 y^2=1$ at the point P...

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Q JEE MAIN_2019_

If the line $x-2 y=12$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $\left(3, \frac{-9}{2}\right)$, then the length of...

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Q JEE MAIN 2019

An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of...

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Q JEE MAIN 2020

If $e_1$ and $e_2$ are the eccentricitis of the ellipse, $\frac{x^2}{18}+\frac{y^2}{4}=1 \quad$ and the hyperbola, $\quad \frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and (...

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Q JEE MAIN 2020

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the...

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Q JEE MAIN 2020

If the co-ordinates of two points $A$ and $B$ are $(\sqrt{7}, 0)$ and $(-\sqrt{7}, 0)$ respectively and $P$ is any...

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Q JEE MAIN 2019

Let $S$ and $S^{\prime}$ be the foci of an ellipse and $B$ be any one of the extremities of its...

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Q JEE MAIN 2020

If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor...

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Q JEE MAIN 2020

Let the line $y=m x$ and the ellipse $2 x^2+y^2=1$ intersect at a point $P$ in the first quadrant. If...

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Q JEE MAIN 2019

If tangents are drawn to the ellipse $x^2+2 y^2=2$ at all points on the ellipse other than its four vertices...

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Q JEE MAIN 2019

The tangent and normal to the ellipse $3 x^2+5 y^2=$ 32 at the point $P(2,2)$ meet the $x$-axis at $Q$...

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Q JEE MAIN 2020

Let $x=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac{1}{2}$. If...

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Q JEE MAIN 2020

The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse...

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Q JEE MAIN 2019

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the...

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Q JEE MAIN 2020

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then...

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Q JEE MAIN 2019

If the tangent to the parabola $\mathrm{y}^2=\mathrm{x}$ at a point $(\alpha, \beta),(\beta>0)$ is also a tangent to the ellipse, $...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

Let $S=\left\{(x, y) \in R^2: \frac{y^2}{1+r}-\frac{x^2}{1-r}=1\right\}$, where where $r \neq \pm 1$. Then $S$ represents

JEE Main Mathematics Medium

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Q JEE Main 2021

The locus of mid-points of the line segments joining $(-3,-5)$ and the points on the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is :

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Q JEE MAIN 2021

If $x^2+9 y^2-4 x+3=0, x, y \in \mathbb{R}$, then $x$ and $y$ respectively lie in the intervals:

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Q JEE MAIN 2021

If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^2}{b^2}+\frac{y^2}{4 a^2}=1$ and the co-ordinate axis...

JEE Main Mathematics Medium

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Q JEE Main 2021

On the ellipse $\frac{x^2}{8}+\frac{y^2}{4}=1$ let $P$ be a point in the second quadrant such that the tangent at $P$ to...

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Q JEE MAIN 2019

If the tangents on the ellipse $4 x^2+y^2=8$ at the points $(1,2)$ and $(\mathrm{a}, \mathrm{b})$ are perpendicular to each other,...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{1}=1$ and the circle $x^2+y^2=3$ at their point...

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Q JEE MAIN_2021_

Let an ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a^2>b^2$, passes through $\left(\sqrt{\frac{3}{2}}, 1\right)$ and has eccentricity $\frac{1}{\sqrt{3}}$. If a circle, centered at focus...

JEE Main Mathematics Easy

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Q JEE MAIN 2021 S2

Let $E$ be an ellipse whose axes are parallel to the co-ordinates axes, having its center at $(3,-4)$, one focus...

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Q JEE MAIN 2021

Let $L$ be a common tangent line to the curves $4 x^2+9 y^2=36$ and $(2 x)^2+(2 y)^2=31$. Then the square...

JEE Main Mathematics Easy

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Q JEE MAIN 2019

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis...

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Q JEE MAIN 2021

A ray of light through $(2,1)$ is reflected at a point $P$ on the $y$-axis and then passes through the...

JEE Main Mathematics Hard

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Q JEE MAIN_2021_

Let $E_1: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$. Let $E_2$ be ahother ellipse such that it touches the end points of major axis of...

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Q JEE MAIN 2021

If a tangent to the ellipse $x^2+4 y^2=4$ meets the tangents at the extremities of its major axis at $B$...

JEE Main Mathematics Medium

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Q JEE-MAIN 2021

Let $L$ be a tangent line to the parabola $y^2=4 x-20$ at $(6,20)$. If $L$ is also a tangent to...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let a tangent be drawn to the ellipse $\frac{x^2}{27}+y^2=1$ at $(3 \sqrt{3} \cos \theta, \sin \theta)$ where $\theta \in\left(0, \frac{\pi}{2}\right)$....

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If the point of intersections of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the circle $x^2+y^2=4 b, b>4$ lie on the curve $...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

The locus of the mid point of the line segment joining the point (4, 3) and the points on the...

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Q JEE MAIN 2022

The line $y=x+1$ meets the ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$ at two points $P$ and $Q$. If $r$ is the radius of the...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

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 $12 m^2$ is equal...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

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...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $P Q$ be a focal chord of the parabola $y^2=4 x$ such that it subtends an angle of $\frac{\pi}{2}$...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

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 $...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

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...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If two tangents drawn from a point ( $\alpha, \beta$ ) lying on the ellipse $25 x^2+4 y^2=1$ to the...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2 x^2-2 y^2=1$. If the ellipse...

JEE Main Mathematics Hard

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Q JEE MAIN 2023

Let a curve $y=f(x), x \in(0, \infty)$ pass through the points $P\left(1, \frac{3}{2}\right)$ and $Q\left(a, \frac{1}{2}\right)$. If the tangent at...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Let the ellipse $3 x^2+p y^2=4$ pass through the centre $C$ of the circle $x^2+y^2-2 x-4 y-11=0$ of radius $r$....

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The centre of a circle $C$ is at the centre of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$. Let $C$ pass through...

JEE Main Mathematics Medium

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Q JEE MAIN_2025

Let $A=\{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R}:|\alpha-1| \leqslant 4$ and $|\beta-5| \leqslant 6\}$ and $...

JEE Main Physics Easy

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Q JEE MAIN_2025

Let the length of a latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be 10 . If its eccentricity is the minimum...

JEE Main Physics Easy

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Q JEE MAIN 2025

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let for two distinct values of p the lines $y=x+\mathrm{p}$ touch the ellipse $\mathrm{E}: \frac{x^2}{4^2}+\frac{y^2}{3^2}=1$ at the points A and...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $C$ be the circle of minimum area enclosing the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{1}{2}$ and foci $...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along $x$-axis. If its...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let the tangent and normal at the point $(3 \sqrt{3}, 1)$ on the ellipse $\frac{x^2}{36}+\frac{y^2}{4}=1$ meet the $y$-axis at the...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Consider ellipses $E_k: k x^2+k^2 y^2=1, k=1,2, \ldots \ldots, 20$. Let $C_k$ be the circle which touches the four chords...

JEE Main Mathematics Hard

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Q JEE MAIN 2023

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 $P Q$...

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Q JEE MAIN 2023

Let the ellipse $E_{\text {in }} x^2+9 y^2=9$ intersect the positive $x$ - and $y$-axes at the points $A$ and...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let the line 2x+3y-k=0,k>0, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let the foci and length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be $( \pm 5,0)$ and $\sqrt{50}$,...

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Q JEE MAIN

If the points of intersection of two distinct conics $x^2+y^2=4 b$ and $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ lie on the curve $y^2=3 x^2$, then...

JEE Main Mathematics Hard

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Q JEE MAIN 2024

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

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 $\left(\frac{5}{2}, \frac{1}{2}\right)$,...

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Q JEE MAIN 2025

If the midpoint of a chord of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ is $(\sqrt{2}, 4 / 3)$, and the length of the...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

At the sea level, the dry air mass percentage composition is given as nitrogen gas: 70.0, oxygen gas: 27.0 and...

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