Question

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, $9x^2+16y^2=144,$ then $PA+PB$ is equal to :

JEE Main 2020 (05 Sep Shift 1)

Options

• A: $16$
• B: $8$
• C: $6$
• D: $9$

Question

If the normal at an end of latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity $\mathrm{e}$ of the ellipse satisfies :

JEE Main 2020 (06 Sep Shift 2)

Options

• A: ${\mathrm{e}}^4+2{\mathrm{e}}^2-1=0$
• B: ${\mathrm{e}}^2+\mathrm{e}-1=0$
• C: ${\mathrm{e}}^4+{\mathrm{e}}^2-1=0$
• D: ${\mathrm{e}}^2+2\mathrm{e}-1=0$

Question

A ray of light through $\left(2,1\right)$ is reflected at a point $P$ on the $y-$ axis and then passes through the point $\left(5,3\right).$ If this reflected ray is the directrix of an ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{\sqrt{53}},$ then the equation of the other directrix can be:

JEE Main 2021 (27 Jul Shift 1)

Options

• A: $11x+7y+8=0$ or $11x+7y-15=0$
• B: $11x-7y-8=0$ or $11x+7y+15=0$
• C: $2x-7y+29=0$ or $2x-7y-7=0$
• D: $2x-7y-39=0$ or $2x-7y-7=0$

Question

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 $F(\alpha ,0),\alpha >0$, of $E$ and radius $\frac{2}{\sqrt{3}}$, intersects $E$ at two points $P$ and $Q$, then $P{Q}^2$ is equal to :

JEE Main 2021 (25 Jul Shift 1)

Options

• A: $\frac{8}{3}$
• B: $\frac{4}{3}$
• C: $\frac{16}{3}$
• D: $3$

Question

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

JEE Main 2021 (31 Aug Shift 2)

Options

• A: $36x^2+16y^2+90x+56y+145=0$
• B: $36x^2+16y^2+108x+80y+145=0$
• C: $9x^2+4y^2+18x+8y+145=0$
• D: $36x^2+16y^2+72x+32y+145=0$

Question

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

JEE Main 2022 (26 Jun Shift 2)

Options

• A: $\frac{\sqrt{3}}{2}$
• B: $\frac{1}{2\sqrt{2}}$
• C: $\frac{1}{\sqrt{2}}$
• D: $\frac{1}{2}$

Question

Let $PQ$ be a focal chord of the parabola $y^2=4x$ such that it subtends an angle of $\frac{\pi }{2}$ at the point $\left(3,0\right)$. Let the line segment $PQ$ 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

JEE Main 2022 (29 Jun Shift 1)

Options

• A: $1+\sqrt{2}$
• B: $3+2\sqrt{2}$
• C: $1+2\sqrt{3}$
• D: $4+5\sqrt{3}$

Question

Let
$S=\left\{\left(x,y\right)\in \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}:9{\left(x-3\right)}^2+16{\left(y-4\right)}^2\le 144\right\}$
and $T=\left\{\left(x,y\right)\in \mathrm{ℝ}\times \mathrm{ℝ}:{\left(x-7\right)}^2+{\left(\mathrm{y}-4\right)}^2\le 36\right\}$
The $n\left(S\cap T\right)$ is equal to ______.

JEE Main 2022 (29 Jul Shift 2)

Enter your answer

Question

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

JEE Main 2023 (12 Apr Shift 1)

Options

• A: $147$
• B: $143$
• C: $137$
• D: $157$

Question

Let $f(x)=x^2+9, g(x)=\frac{x}{x-9}$ and $\mathrm{a}=f \circ g(10), \mathrm{b}=g \circ f(3)$. If $\mathrm{e}$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{a}+\frac{y^2}{b}=1$, then $8 \mathrm{e}^2+l^2$ is equal to.

JEE Main 2024 (09 Apr Shift 1)

Options

• A: 8
• B: 16
• C: 6
• D: 12

Question

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 $P \text{and} Q$ 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:

JEE Main 2024 (01 Feb Shift 2)

Options

• A: $\frac{11}{19}$
• B: $\frac{13}{21}$
• C: $\frac{\sqrt{139}}{23}$
• D: $\frac{\sqrt{13}}{7}$

Question

Let the line $2 x+3 y-\mathrm{k}=0, \mathrm{k}>0$, intersect the $x$-axis and $y$-axis at the points $\mathrm{A}$ and $\mathrm{B}$, respectively. If the equation of the circle having the line segment $\mathrm{AB}$ as a diameter is $x^2+y^2-3 x-2 y=0$ and the length of the latus rectum of the ellipse $x^2+9 y^2=k^2$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $2 \mathrm{~m}+\mathrm{n}$ is equal to

JEE Main 2024 (05 Apr Shift 1)

Options

• A: 11
• B: 10
• C: 12
• D: 13

Question

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

JEE Main 2024 (27 Jan Shift 1)

Options

• A: $\frac{\sqrt{1691}}{5}$
• B: $\frac{\sqrt{2009}}{5}$
• C: $\frac{\sqrt{1741}}{5}$
• D: $\frac{\sqrt{1541}}{5}$

Question

Let $A(\alpha ,0)$ and $B(0,\beta )$ be the points on the line $5x+7y=50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7: 3$. Let $3x-25=0$ be a directrix of the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x-$axis passes through $P$, then the length of the latus rectum of $E$ is equal to

JEE Main 2024 (30 Jan Shift 2)

Options

• A: $\frac{25}{3}$
• B: $\frac{32}{9}$
• C: $\frac{25}{9}$
• D: $\frac{32}{5}$