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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 :
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.
${\pi \over 2} + {\tan ^{ - 1}}\left( {{1 \over 4}} \right)$
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?
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 :
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 :
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 :
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.
$18$
2014 JEE Mains MCQ
JEE Main 2014 (Offline)
The locus of the foot of perpendicular drawn from the centre of the ellipse ${x^2} + 3{y^2} = 6$ on any tangent to it is :
A.
$\left( {{x^2} + {y^2}} \right) ^2 = 6{x^2} + 2{y^2}$
B.
$\left( {{x^2} + {y^2}} \right) ^2 = 6{x^2} - 2{y^2}$
C.
$\left( {{x^2} - {y^2}} \right) ^2 = 6{x^2} + 2{y^2}$
D.
$\left( {{x^2} - {y^2}} \right) ^2 = 6{x^2} - 2{y^2}$
2013 JEE Mains MCQ
JEE Main 2013 (Offline)
The equation of the circle passing through the foci of the ellipse ${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$, and having centre at $(0,3)$ is :
A.
${x^2} + {y^2} - 6y - 7 = 0$
B.
${x^2} + {y^2} - 6y + 7 = 0$
C.
${x^2} + {y^2} - 6y - 5 = 0$
D.
${x^2} + {y^2} - 6y + 5 = 0$
2012 JEE Mains MCQ
AIEEE 2012
STATEMENT-1 : An equation of a common tangent to the parabola ${y^2} = 16\sqrt 3 x$ and the ellipse $2{x^2} + {y^2} = 4$ is $y = 2x + 2\sqrt 3 $

STATEMENT-2 :If line $y = mx + {{4\sqrt 3 } \over m},\left( {m \ne 0} \right)$ is a common tangent to the parabola ${y^2} = 16\sqrt {3x} $and the ellipse $2{x^2} + {y^2} = 4$, then $m$ satisfies ${m^4} + 2{m^2} = 24$

A.
Statement-1 is false, Statement-2 is true.
B.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
C.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
D.
Statement-1 is true, Statement-2 is false.
2012 JEE Mains MCQ
AIEEE 2012
An ellipse is drawn by taking a diameter of thec circle ${\left( {x - 1} \right)^2} + {y^2} = 1$ as its semi-minor axis and a diameter of the circle ${x^2} + {\left( {y - 2} \right)^2} = 4$ is semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is :
A.
$4{x^2} + {y^2} = 4$
B.
${x^2} + 4{y^2} = 8$
C.
$4{x^2} + {y^2} = 8$
D.
${x^2} + 4{y^2} = 16$
2011 JEE Mains MCQ
AIEEE 2011
Equation of the ellipse whose axes of coordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt {{2 \over 5}} $ is :
A.
$5{x^2} + 3{y^2} - 48 = 0$
B.
$3{x^2} + 5{y^2} - 15 = 0$
C.
$5{x^2} + 3{y^2} - 32 = 0$
D.
$3{x^2} + 5{y^2} - 32 = 0$
2009 JEE Mains MCQ
AIEEE 2009
The ellipse ${x^2} + 4{y^2} = 4$ is inscribed in a rectangle aligned with the coordinate axex, which in turn is inscribed in another ellipse that passes through the point $(4,0)$. Then the equation of the ellipse is :
A.
${x^2} + 12{y^2} = 16$
B.
$4{x^2} + 48{y^2} = 48$
C.
$4{x^2} + 64{y^2} = 48$
D.
${x^2} + 16{y^2} = 16$
2008 JEE Mains MCQ
AIEEE 2008
A focus of an ellipse is at the origin. The directrix is the line $x=4$ and the eccentricity is ${{1 \over 2}}$. Then the length of the semi-major axis is :
A.
${{8 \over 3}}$
B.
${{2 \over 3}}$
C.
${{4 \over 3}}$
D.
${{5 \over 3}}$
2006 JEE Mains MCQ
AIEEE 2006
In the ellipse, the distance between its foci is $6$ and minor axis is $8$. Then its eccentricity is :
A.
${3 \over 5}$
B.
${1 \over 2}$
C.
${4 \over 5}$
D.
${1 \over {\sqrt 5 }}$
2005 JEE Mains MCQ
AIEEE 2005
An ellipse has $OB$ as semi minor axis, $F$ and $F$' its focii and theangle $FBF$' is a right angle. Then the eccentricity of the ellipse is :
A.
${1 \over {\sqrt 2 }}$
B.
${1 \over 2}$
C.
${1 \over 4}$
D.
${1 \over {\sqrt 3 }}$
2004 JEE Mains MCQ
AIEEE 2004
The eccentricity of an ellipse, with its centre at the origin, is ${1 \over 2}$. If one of the directrices is $x=4$, then the equation of the ellipse is :
A.
$4{x^2} + 3{y^2} = 1$
B.
$3{x^2} + 4{y^2} = 12$
C.
$4{x^2} + 3{y^2} = 12$
D.
$3{x^2} + 4{y^2} = 1$
2026 JEE Mains Numerical
JEE Main 2026 (Online) 24th January Evening Shift

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 eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to $\_\_\_\_$ .

2025 JEE Mains Numerical
JEE Main 2025 (Online) 28th January Morning Shift

Let $\mathrm{E}_1: \frac{x^2}{9}+\frac{y^2}{4}=1$ be an ellipse. Ellipses $\mathrm{E}_{\mathrm{i}}$ 's are constructed such that their centres and eccentricities are same as that of $\mathrm{E}_1$, and the length of minor axis of $\mathrm{E}_{\mathrm{i}}$ is the length of major axis of $E_{i+1}(i \geq 1)$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi}\left(\sum\limits_{i=1}^{\infty} A_i\right)$, is equal to _______.