2022
JEE Mains
MCQ
JEE Main 2022 (Online) 27th June Morning Shift
Let the eccentricity of an ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, $a > b$, be ${1 \over 4}$. If this ellipse passes through the point $\left( { - 4\sqrt {{2 \over 5}} ,3} \right)$, then ${a^2} + {b^2}$ is equal to :
A.
29
B.
31
C.
32
D.
34
2022
JEE Mains
MCQ
JEE Main 2022 (Online) 26th June Evening Shift
If m is the slope of a common tangent to the curves ${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$ and ${x^2} + {y^2} = 12$, then $12{m^2}$ is equal to :
A.
6
B.
9
C.
10
D.
12
2022
JEE Mains
MCQ
JEE Main 2022 (Online) 26th June Evening Shift
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 :
A.
${{\sqrt 3 } \over 2}$
B.
${1 \over {2\sqrt 2 }}$
C.
${1 \over {\sqrt 2 }}$
D.
${1 \over 2}$
2022
JEE Mains
MCQ
JEE Main 2022 (Online) 25th June Evening Shift
The line y = x + 1 meets the ellipse ${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$ at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)2 is equal to :
A.
20
B.
12
C.
11
D.
8
2022
JEE Mains
MCQ
JEE Main 2022 (Online) 24th June Evening Shift
Let the maximum area of the triangle that can be inscribed in the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2$, 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 :
A.
${{\sqrt 3 } \over 2}$
B.
${1 \over 2}$
C.
${1 \over {\sqrt 2 }}$
D.
${{\sqrt 3 } \over 4}$
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 28th July Evening Shift
Let the tangents at the points $\mathrm{P}$ and $\mathrm{Q}$ on the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$ meet at the point $R(\sqrt{2}, 2 \sqrt{2}-2)$. If $\mathrm{S}$ is the focus of the ellipse on its negative major axis, then $\mathrm{SP}^{2}+\mathrm{SQ}^{2}$ is equal to ___________.
Correct Answer: 13
Explanation:
$E \equiv {{{x^2}} \over 2} + {{{y^2}} \over 4} = 1$
$\eqalign{ & T \equiv y = mx\, \pm \,\sqrt {2{m^2} + 4} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \downarrow \left( {\sqrt 2 ,2\sqrt 2 - 2} \right) \cr} $
$ \Rightarrow \left( {2\sqrt 2 - 2 - m\sqrt 2 } \right) = \pm \,\sqrt {2{m^2} + 4} $
$ \Rightarrow 2{m^2} - 2m\sqrt 2 \left( {2\sqrt {2 - 2} } \right) + 4(3 - 2\sqrt 2 ) = 2{m^2} + 4$
$ \Rightarrow - 2\sqrt 2 m(2\sqrt 2 - 2) = 4 - 12 + 8\sqrt 2 $
$ \Rightarrow - 4\sqrt 2 m(\sqrt 2 - 1) = 8(\sqrt 2 - 1)$
$ \Rightarrow m = - \sqrt 2 $ and $m \to \infty $
$\therefore$ Tangents are $x = \sqrt 2 $ and $y = - \sqrt 2 x + \sqrt 8 $
$\therefore$ $P(\sqrt 2 ,0)$ and $Q(1,\sqrt 2 )$
and $S = (0, - \sqrt 2 )$
$\therefore$ ${(PS)^2} + {(QS)^2} = 4 + 9 = 13$
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 27th July Morning Shift
If the length of the latus rectum of the ellipse $x^{2}+4 y^{2}+2 x+8 y-\lambda=0$ is 4 , and $l$ is the length of its major axis, then $\lambda+l$ is equal to ____________.
Correct Answer: 75
Explanation:
Equation of ellipse is : ${x^2} + 4{y^2} + 2x + 8y - \lambda = 0$
${(x + 1)^2} + 4{(y + 1)^2} = \lambda + 5$
${{{{(x + 1)}^2}} \over {\lambda + 5}} + {{{{(y + 1)}^2}} \over {\left( {{{\lambda + 5} \over 4}} \right)}} = 1$
Length of latus rectum $ = {{2\,.\,\left( {{{\lambda + 5} \over 4}} \right)} \over {\sqrt {\lambda + 5} }} = 4$.
$\therefore$ $\lambda = 59$.
Length of major axis $ = 2\,.\,\sqrt {\lambda + 5} = 16 = l$
$\therefore$ $\lambda + l = 75$.
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 24th June Morning Shift
If two tangents drawn from a point ($\alpha$, $\beta$) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other, then the value of (10$\alpha$ + 5)2 + (16$\beta$2 + 50)2 equals ___________.
Correct Answer: 2929
Explanation:
$\because(\alpha, \beta)$ lies on the given ellipse, $25 \alpha^{2}+4 \beta^{2}=1\quad\quad...(i)$
Tangent to the parabola, $y=m x+\frac{1}{m}$ passes through $(\alpha, \beta)$. So, $\alpha m^{2}-\beta m+1=0$ has roots $m_{1}$ and $4 m_{1}$,
$ m_{1}+4 m_{1}=\frac{\beta}{\alpha} \text { and } m_{1} \cdot 4 m_{1}=\frac{1}{\alpha} $
Gives that $4 \beta^{2}=25 \alpha \quad\quad...(ii)$
from (i) and (ii)
$25\left(\alpha^{2}+\alpha\right)=1\quad\quad...(iii)$
Now, $(10 \alpha+5)^{2}+\left(16 \beta^{2}+50\right)^{2}$
$=25(2 \alpha+1)^{2}+2500(2 \alpha+1)^{2}$
$=2525\left(4 \alpha^{2}+4 \alpha+1\right)$ from equation (iii)
$=2525\left(\frac{4}{25}+1\right)$
$=2929$
Tangent to the parabola, $y=m x+\frac{1}{m}$ passes through $(\alpha, \beta)$. So, $\alpha m^{2}-\beta m+1=0$ has roots $m_{1}$ and $4 m_{1}$,
$ m_{1}+4 m_{1}=\frac{\beta}{\alpha} \text { and } m_{1} \cdot 4 m_{1}=\frac{1}{\alpha} $
Gives that $4 \beta^{2}=25 \alpha \quad\quad...(ii)$
from (i) and (ii)
$25\left(\alpha^{2}+\alpha\right)=1\quad\quad...(iii)$
Now, $(10 \alpha+5)^{2}+\left(16 \beta^{2}+50\right)^{2}$
$=25(2 \alpha+1)^{2}+2500(2 \alpha+1)^{2}$
$=2525\left(4 \alpha^{2}+4 \alpha+1\right)$ from equation (iii)
$=2525\left(\frac{4}{25}+1\right)$
$=2929$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 1st September Evening Shift
Let $\theta$ be the acute angle between the tangents to the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 1} = 1$ and the circle ${x^2} + {y^2} = 3$ at their point of intersection in the first quadrant. Then tan$\theta$ is equal to :
A.
${5 \over {2\sqrt 3 }}$
B.
${2 \over {\sqrt 3 }}$
C.
${4 \over {\sqrt 3 }}$
D.
2
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 31st August Evening Shift
The locus of mid-points of the line segments joining ($-$3, $-$5) and the points on the ellipse ${{{x^2}} \over 4} + {{{y^2}} \over 9} = 1$ is :
A.
$9{x^2} + 4{y^2} + 18x + 8y + 145 = 0$
B.
$36{x^2} + 16{y^2} + 90x + 56y + 145 = 0$
C.
$36{x^2} + 16{y^2} + 108x + 80y + 145 = 0$
D.
$36{x^2} + 16{y^2} + 72x + 32y + 145 = 0$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 31st August Evening Shift
An angle of intersection of the curves, ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ and x2 + y2 = ab, a > b, is :
A.
${\tan ^{ - 1}}\left( {{{a + b} \over {\sqrt {ab} }}} \right)$
B.
${\tan ^{ - 1}}\left( {{{a - b} \over {2\sqrt {ab} }}} \right)$
C.
${\tan ^{ - 1}}\left( {{{a - b} \over {\sqrt {ab} }}} \right)$
D.
${\tan ^{ - 1}}\left( {2\sqrt {ab} } \right)$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 31st August Morning Shift
The line $12x\cos \theta + 5y\sin \theta = 60$ is tangent to which of the following curves?
A.
x2 + y2 = 169
B.
144x2 + 25y2 = 3600
C.
25x2 + 12y2 = 3600
D.
x2 + y2 = 60
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th August Morning Shift
If x2 + 9y2 $-$ 4x + 3 = 0, x, y $\in$ R, then x and y respectively lie in the intervals :
A.
$\left[ { - {1 \over 3},{1 \over 3}} \right]$ and $\left[ { - {1 \over 3},{1 \over 3}} \right]$
B.
$\left[ { - {1 \over 3},{1 \over 3}} \right]$ and [1, 3]
C.
[1, 3] and [1, 3]
D.
[1, 3] and $\left[ { - {1 \over 3},{1 \over 3}} \right]$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th August Morning Shift
On the ellipse ${{{x^2}} \over 8} + {{{y^2}} \over 4} = 1$ let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S' be the foci of the ellipse and e be its eccentricity. If A is the area of the triangle SPS' then, the value of (5 $-$ e2). A is :
A.
6
B.
12
C.
14
D.
24
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th July Morning Shift
A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity ${1 \over 3}$ and the distance of the nearer focus from this directrix is ${8 \over {\sqrt {53} }}$, then the equation of the other directrix can be :
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
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th July Evening Shift
If a tangent to the ellipse x2 + 4y2 = 4 meets the tangents at the extremities of it major axis at B and C, then the circle with BC as diameter passes through the point :
A.
$(\sqrt 3 ,0)$
B.
$(\sqrt 2 ,0)$
C.
(1, 1)
D.
($-$1, 1)
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th July Morning Shift
Let an ellipse $E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ${a^2} > {b^2}$, passes through $\left( {\sqrt {{3 \over 2}} ,1} \right)$ and has eccentricity ${1 \over {\sqrt 3 }}$. If a circle, centered at focus F($\alpha$, 0), $\alpha$ > 0, of E and radius ${2 \over {\sqrt 3 }}$, intersects E at two points P and Q, then PQ2 is equal to :
A.
${8 \over 3}$
B.
${4 \over 3}$
C.
${{16} \over 3}$
D.
3
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 22th July Evening Shift
Let ${E_1}:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,a > b$. Let E2 be another ellipse such that it touches the end points of major axis of E1 and the foci of E2 are the end points of minor axis of E1. If E1 and E2 have same eccentricities, then its value is :
A.
${{ - 1 + \sqrt 5 } \over 2}$
B.
${{ - 1 + \sqrt 8 } \over 2}$
C.
${{ - 1 + \sqrt 3 } \over 2}$
D.
${{ - 1 + \sqrt 6 } \over 2}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 18th March Evening Shift
Let a tangent be drawn to the ellipse ${{{x^2}} \over {27}} + {y^2} = 1$ at $(3\sqrt 3 \cos \theta ,\sin \theta )$ where $0 \in \left( {0,{\pi \over 2}} \right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :
A.
${{\pi \over 6}}$
B.
${{\pi \over 3}}$
C.
${{\pi \over 8}}$
D.
${{\pi \over 4}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 16th March Evening Shift
If the points of intersections of the ellipse ${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$ and the
circle x2 + y2 = 4b, b > 4 lie on the curve y2 = 3x2, then b is equal to :
circle x2 + y2 = 4b, b > 4 lie on the curve y2 = 3x2, then b is equal to :
A.
12
B.
10
C.
6
D.
5
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th February Evening Shift
If the curve x2 + 2y2 = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is :
A.
${\pi \over 2} - {\tan ^{ - 1}}\left( {{1 \over 4}} \right)$
B.
${\pi \over 2} + {\tan ^{ - 1}}\left( {{1 \over 3}} \right)$
C.
${\pi \over 2} - {\tan ^{ - 1}}\left( {{1 \over 3}} \right)$
D.
${\pi \over 2} + {\tan ^{ - 1}}\left( {{1 \over 4}} \right)$
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 27th August Morning Shift
If the minimum area of the triangle formed by a tangent to the ellipse ${{{x^2}} \over {{b^2}}} + {{{y^2}} \over {4{a^2}}} = 1$ and the co-ordinate axis is kab, then k is equal to _______________.
Correct Answer: 2
Explanation:
Tangent
${{x\cos \theta } \over b} + {{y\sin \theta } \over {2a}} = 1$
So, area $(\Delta OAB) = {1 \over 2} \times {b \over {\cos \theta }} \times {{2a} \over {\sin \theta }}$
$ = {{2ab} \over {\sin 2\theta }} \ge 2ab$
$\Rightarrow$ k = 2
${{x\cos \theta } \over b} + {{y\sin \theta } \over {2a}} = 1$
So, area $(\Delta OAB) = {1 \over 2} \times {b \over {\cos \theta }} \times {{2a} \over {\sin \theta }}$
$ = {{2ab} \over {\sin 2\theta }} \ge 2ab$
$\Rightarrow$ k = 2
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 27th July Evening Shift
Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3, $-$4), one focus at (4, $-$4) and one vertex at (5, $-$4). If mx $-$ y = 4, m > 0 is a tangent to the ellipse E, then the value of 5m2 is equal to _____________.
Correct Answer: 3
Explanation:
Given C(3, $-$4), S(4, $-$4)
and A(5, $-$4)
Hence, a = 2 & ae = 1
$\Rightarrow$ e = ${1 \over 2}$
$\Rightarrow$ b2 = 3
So, $E:{{{{(x - 3)}^2}} \over 4} + {{{{(y + 4)}^2}} \over 3} = 1$
Intersecting with given tangent.
${{{x^2} - 6x + 9} \over 4} + {{{m^2}{x^2}} \over 3} = 1$
Now, D = 0 (as it is tngent)
So, 5m2 = 3.
and A(5, $-$4)
Hence, a = 2 & ae = 1
$\Rightarrow$ e = ${1 \over 2}$
$\Rightarrow$ b2 = 3
So, $E:{{{{(x - 3)}^2}} \over 4} + {{{{(y + 4)}^2}} \over 3} = 1$
Intersecting with given tangent.
${{{x^2} - 6x + 9} \over 4} + {{{m^2}{x^2}} \over 3} = 1$
Now, D = 0 (as it is tngent)
So, 5m2 = 3.
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 26th February Evening Shift
Let L be a common tangent line to the curves
4x2 + 9y2 = 36 and (2x)2 + (2y)2 = 31. Then the
square of the slope of the line L is __________.
4x2 + 9y2 = 36 and (2x)2 + (2y)2 = 31. Then the
square of the slope of the line L is __________.
Correct Answer: 3
Explanation:
Tangent to the curve ${{{x^2}} \over 9} + {{{y^2}} \over {14}} = 1$ is
$y = mx + \sqrt {9{m^2} + 4} $
and equation of tangent to the curve ${x^2} + {y^2} = {{31} \over 4}$ is
$y = mx + \sqrt {{{31} \over 4}{{(1 + m)}^2}} $
for common tangent $9{m^2} + 4 = {{31} \over 4} + {{31} \over 4}{m^2}$
$ \Rightarrow {5 \over 4}{m^2} = {{15} \over 4}$
$ \Rightarrow {m^2} = 3$
$y = mx + \sqrt {9{m^2} + 4} $
and equation of tangent to the curve ${x^2} + {y^2} = {{31} \over 4}$ is
$y = mx + \sqrt {{{31} \over 4}{{(1 + m)}^2}} $
for common tangent $9{m^2} + 4 = {{31} \over 4} + {{31} \over 4}{m^2}$
$ \Rightarrow {5 \over 4}{m^2} = {{15} \over 4}$
$ \Rightarrow {m^2} = 3$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Evening Slot
If the normal at an end of a latus rectum of an
ellipse passes through an extremity of the
minor axis, then the eccentricity e of the ellipse
satisfies :
A.
e4 + 2e2 – 1 = 0
B.
e4 + e2 – 1 = 0
C.
e2 + 2e – 1 = 0
D.
e2 + e – 1 = 0
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Morning Slot
Which of the following points lies on the locus of the foot of perpedicular drawn upon any tangent
to the ellipse,
${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$
from any of its foci?
${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$
from any of its foci?
A.
$\left( { - 1,\sqrt 3 } \right)$
B.
$\left( { - 2,\sqrt 3 } \right)$
C.
$\left( { - 1,\sqrt 2 } \right)$
D.
$\left( {1,2 } \right)$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 5th September Morning Slot
If the co-ordinates of two points A and B
are $\left( {\sqrt 7 ,0} \right)$ and $\left( { - \sqrt 7 ,0} \right)$ respectively and
P is any point on the conic, 9x2 + 16y2 = 144, then PA + PB is equal to :
are $\left( {\sqrt 7 ,0} \right)$ and $\left( { - \sqrt 7 ,0} \right)$ respectively and
P is any point on the conic, 9x2 + 16y2 = 144, then PA + PB is equal to :
A.
8
B.
9
C.
16
D.
6
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 4th September Evening Slot
Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is ${1 \over 2}$. If P(1, $\beta $), $\beta $ > 0 is a point on this ellipse, then the equation of the normal to it at P is :
A.
4x – 3y = 2
B.
8x – 2y = 5
C.
7x – 4y = 1
D.
4x – 2y = 1
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 4th September Morning Slot
Let ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function,
$\phi \left( t \right) = {5 \over {12}} + t - {t^2}$, then a2 + b2 is equal to :
$\phi \left( t \right) = {5 \over {12}} + t - {t^2}$, then a2 + b2 is equal to :
A.
145
B.
126
C.
135
D.
116
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 9th January Evening Slot
The length of the minor axis (along y-axis) of
an ellipse in the standard form is ${4 \over {\sqrt 3 }}$. If this
ellipse touches the line, x + 6y = 8; then its
eccentricity is :
A.
${1 \over 3}\sqrt {{{11} \over 3}} $
B.
${1 \over 2}\sqrt {{5 \over 3}} $
C.
$\sqrt {{5 \over 6}} $
D.
${1 \over 2}\sqrt {{{11} \over 3}} $
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 8th January Morning Slot
Let the line y = mx and the ellipse 2x2 + y2 = 1
intersect at a ponit P in the first quadrant. If the
normal to this ellipse at P meets the co-ordinate axes at $\left( { - {1 \over {3\sqrt 2 }},0} \right)$ and (0, $\beta $), then $\beta $ is equal
to :
A.
${{\sqrt 2 } \over 3}$
B.
${2 \over 3}$
C.
${{2\sqrt 2 } \over 3}$
D.
${2 \over {\sqrt 3 }}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 7th January Evening Slot
If 3x + 4y = 12$\sqrt 2 $ is a tangent to the ellipse
${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1$ for some $a$ $ \in $ R, then the distance between the foci of the ellipse is :
${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1$ for some $a$ $ \in $ R, then the distance between the foci of the ellipse is :
A.
$2\sqrt 5 $
B.
$2\sqrt 7 $
C.
4
D.
$2\sqrt 2 $
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 7th January Morning Slot
If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12,
then the length of its latus rectum is :
A.
$\sqrt 3 $
B.
$3\sqrt 2 $
C.
${3 \over {\sqrt 2 }}$
D.
$2\sqrt 3 $
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th April Evening Slot
An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of the following points?
A.
$\left( {2,\sqrt 2 } \right)$
B.
$\left( {2,2\sqrt 2 } \right)$
C.
$\left( {\sqrt 2 ,2} \right)$
D.
$\left( {1,2\sqrt 2 } \right)$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th April Morning Slot
If the normal to the ellipse 3x2
+ 4y2
= 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent
to the ellipse at P passes through Q(4,4) then PQ is equal to :
A.
${{\sqrt {61} } \over 2}$
B.
${{\sqrt {221} } \over 2}$
C.
${{\sqrt {157} } \over 2}$
D.
${{5\sqrt 5 } \over 2}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th April Evening Slot
The tangent and normal to the ellipse 3x2
+ 5y2
= 32 at the point P(2, 2) meet the x-axis at Q and R,
respectively. Then the area (in sq. units) of the triangle PQR is :
A.
${{14} \over 3}$
B.
${{16} \over 3}$
C.
${{68} \over {15}}$
D.
${{34} \over {15}}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th April Morning Slot
If the line x – 2y = 12 is tangent to the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ at the point $\left( {3, - {9 \over 2}} \right)$ , then the length of the latus
rectum of the ellipse is :
A.
5
B.
9
C.
$8\sqrt 3 $
D.
$12\sqrt 2 $
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th April Evening Slot
If the tangent to the parabola y2 = x at a point
($\alpha $, $\beta $), ($\beta $ > 0) is also a tangent to the ellipse,
x2 + 2y2 = 1, then $\alpha $ is equal to :
A.
$\sqrt 2 + 1$
B.
$\sqrt 2 - 1$
C.
$2\sqrt 2 + 1$
D.
$2\sqrt 2 - 1$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 8th April Evening Slot
In an ellipse, with centre at the origin, if the
difference of the lengths of major axis and minor
axis is 10 and one of the foci is at (0,5$\sqrt 3$), then
the length of its latus rectum is :
A.
5
B.
8
C.
10
D.
6
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 8th April Morning Slot
If the tangents on the ellipse 4x2 + y2 = 8 at the
points (1, 2) and (a, b) are perpendicular to each
other, then a2 is equal to :
A.
${{2} \over {17}}$
B.
${{64} \over {17}}$
C.
${{128} \over {17}}$
D.
${{4} \over {17}}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th January Evening Slot
Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $\Delta $S'BS is a right angled triangle with right angle at B and area ($\Delta $S'BS) = 8 sq. units, then the length of a latus rectum of the ellipse is :
A.
2
B.
4$\sqrt 2 $
C.
4
D.
2$\sqrt 2 $
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Evening Slot
Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?
A.
$\left( {4\sqrt 2 ,2\sqrt 3 } \right)$
B.
$\left( {4\sqrt 3 ,2\sqrt 3 } \right)$
C.
$\left( {4\sqrt 3 ,2\sqrt 2 } \right)$
D.
$\left( {4\sqrt 2 ,2\sqrt 2 } \right)$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Morning Slot
If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve :
A.
${{{x^2}} \over 2} + {{{y^2}} \over 4} = 1$
B.
${1 \over {2{x^2}}} + {1 \over {4{y^2}}} = 1$
C.
${1 \over {4{x^2}}} + {1 \over {2{y^2}}} = 1$
D.
${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th January Evening Slot
Let S = $\left\{ {\left( {x,y} \right) \in {R^2}:{{{y^2}} \over {1 + r}} - {{{x^2}} \over {1 - r}}} \right\};r \ne \pm 1.$ Then S represents :
A.
an ellipse whose eccentricity is ${1 \over {\sqrt {r + 1} }},$ where r > 1
B.
an ellipse whose eccentricity is ${2 \over {\sqrt {r + 1} }},$ where 0 < r < 1
C.
an ellipse whose eccentricity is ${2 \over {\sqrt {r - 1} }},$ where 0 < r < 1
D.
an ellipse whose eccentricity is $\sqrt {{2 \over {r + 1}}}$, where r > 1
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 16th April Morning Slot
If the length of the latus rectum of an ellipse is 4 units and the distance between a focus an its nearest vertex on the major axis is ${3 \over 2}$ units, then its eccentricity is :
A.
${1 \over 2}$
B.
${1 \over 3}$
C.
${2 \over 3}$
D.
${1 \over 9}$
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 9th April Morning Slot
The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate
axes and passing through the points (4, −1) and (−2, 2) is :
A.
${1 \over 2}$
B.
${2 \over {\sqrt 5 }}$
C.
${{\sqrt 3 } \over 2}$
D.
${{\sqrt 3 } \over 4}$
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 8th April Morning Slot
Consider an ellipse, whose center is at the origin and its major axis is along the x-axis. If its eccentricity is ${3 \over 5}$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilatateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is :
A.
8
B.
32
C.
80
D.
40
2017
JEE Mains
MCQ
JEE Main 2017 (Offline)
The eccentricity of an ellipse whose centre is at the origin is ${1 \over 2}$. If one of its directrices is x = – 4, then the
equation of the normal to it at $\left( {1,{3 \over 2}} \right)$ is :
A.
2y – x = 2
B.
4x – 2y = 1
C.
4x + 2y = 7
D.
x + 2y = 4
2016
JEE Mains
MCQ
JEE Main 2016 (Online) 9th April Morning Slot
If the tangent at a point on the ellipse ${{{x^2}} \over {27}} + {{{y^2}} \over 3} = 1$ meets the coordinate axes at A and B, and O is the origin, then the
minimum area (in sq. units) of the triangle OAB is :
A.
${9 \over 2}$
B.
$3\sqrt 3 $
C.
$9\sqrt 3 $
D.
9
2015
JEE Mains
MCQ
JEE Main 2015 (Offline)
The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 5} = 1$, is :
A.
${{27 \over 2}}$
B.
$27$
C.
${{27 \over 4}}$
D.
$18$
