# Ellipse · Mathematics · JEE Main

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JEE Main 2024 (Online) 9th April Morning Shift
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 ...
JEE Main 2024 (Online) 5th April Morning Shift
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 $$\mathrm{B}$$, respecti...
JEE Main 2024 (Online) 1st February Evening Shift
Let $\mathrm{P}$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let the line passing through $\mathrm{P}$ and parallel to $y$-axis meet th...
JEE Main 2024 (Online) 1st February Morning Shift
Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \mathrm{a}>\mathrm{b}$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latusre...
JEE Main 2024 (Online) 31st January Evening Shift
Let $$P$$ be a parabola with vertex $$(2,3)$$ and directrix $$2 x+y=6$$. Let an ellipse $$E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$$, of eccentricity...
JEE Main 2024 (Online) 30th January Evening Shift
Let $$A(\alpha, 0)$$ and $$B(0, \beta)$$ be the points on the line $$5 x+7 y=50$$. Let the point $$P$$ divide the line segment $$A B$$ internally in t...
JEE Main 2024 (Online) 30th January Morning Shift
If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :
JEE Main 2024 (Online) 27th January Morning Shift
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 :
JEE Main 2023 (Online) 13th April Morning Shift
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 points $$... JEE Main 2023 (Online) 12th April Morning Shift Let$$\mathrm{P}\left(\frac{2 \sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}}\right), \mathrm{Q}, \mathrm{R}$$and$$\mathrm{S}$$be four points on the ellips... JEE Main 2023 (Online) 11th April Evening Shift If the radius of the largest circle with centre (2,0) inscribed in the ellipse$$x^2+4y^2=36$$is r, then 12r$$^2$$is equal to : JEE Main 2023 (Online) 11th April Morning Shift Consider ellipses$$\mathrm{E}_{k}: k x^{2}+k^{2} y^{2}=1, k=1,2, \ldots, 20$$. Let$$\mathrm{C}_{k}$$be the circle which touches the four chords joi... JEE Main 2023 (Online) 10th April Evening Shift Let a circle of radius 4 be concentric to the ellipse$$15 x^{2}+19 y^{2}=285$$. Then the common tangents are inclined to the minor axis of the ellips... JEE Main 2023 (Online) 10th April Morning Shift Let the ellipse$$E:{x^2} + 9{y^2} = 9$$intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter... JEE Main 2023 (Online) 6th April Evening Shift In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who spe... JEE Main 2023 (Online) 31st January Morning Shift If the maximum distance of normal to the ellipse$$\frac{x^{2}}{4}+\frac{y^{2}}{b^{2}}=1, b ...
JEE Main 2022 (Online) 29th July Morning Shift
Let a line L pass through the point of intersection of the lines $$b x+10 y-8=0$$ and $$2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right... JEE Main 2022 (Online) 26th July Evening Shift The acute angle between the pair of tangents drawn to the ellipse$$2 x^{2}+3 y^{2}=5$$from the point$$(1,3)$$is : JEE Main 2022 (Online) 25th July Evening Shift If the ellipse$$\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$$-axis and the line$$\f...
JEE Main 2022 (Online) 30th June Morning Shift
Let the eccentricity of the ellipse $${x^2} + {a^2}{y^2} = 25{a^2}$$ be b times the eccentricity of the hyperbola $${x^2} - {a^2}{y^2} = 5$$, where a ...
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 th...
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 e...
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 eccent...
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 ...
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 i...
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 $${x^2} + {y^2} = 3... JEE Main 2021 (Online) 31st August Evening Shift The locus of mid-points of the line segments joining ($$-$$3,$$-$$5) and the points on the ellipse$${{{x^2}} \over 4} + {{{y^2}} \over 9} = 1$$is : JEE Main 2021 (Online) 31st August Evening Shift An angle of intersection of the curves,$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$and x2 + y2 = ab, a > b, is :... JEE Main 2021 (Online) 31st August Morning Shift The line$$12x\cos \theta + 5y\sin \theta = 60$$is tangent to which of the following curves? JEE Main 2021 (Online) 27th August Morning Shift If x2 + 9y2$$-$$4x + 3 = 0, x, y$$\in$$R, then x and y respectively lie in the intervals : JEE Main 2021 (Online) 26th August Morning Shift On the ellipse$${{{x^2}} \over 8} + {{{y^2}} \over 4} = 1$$let P be a point in the second quadrant such that the tangent at P to the ellipse is perp... JEE Main 2021 (Online) 27th July Morning Shift A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix... JEE Main 2021 (Online) 25th July Evening Shift If a tangent to the ellipse x2 + 4y2 = 4 meets the tangents at the extremities of it major axis at B and C, then the circle with BC as diameter passes... JEE Main 2021 (Online) 25th July Morning Shift Let an ellipse$$E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$,$${a^2} > {b^2}$$, passes through$$\left( {\sqrt {{3 \over 2}} ,1} \ri...
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 a...
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 $$0 \in \left( {0,{\pi \... JEE Main 2021 (Online) 16th March Evening Shift If the points of intersections of the ellipse$${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$$and the circle x2 + y2 = 4b, b > 4 lie on the ... JEE Main 2021 (Online) 25th February Evening Shift If the curve x2 + 2y2 = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is :... JEE Main 2020 (Online) 6th September Evening Slot If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfie... JEE Main 2020 (Online) 6th September Morning Slot Which of the following points lies on the locus of the foot of perpedicular drawn upon any tangent to the ellipse,$${{{x^2}} \over 4} + {{{y^2}} \ove...
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 o...
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 i...
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 i...
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 =...
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-ordinat...
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 dista...
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 :
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?
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 throug...
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 ...
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 $$\left( {3, - {9 \over 2}} \... JEE Main 2019 (Online) 9th April Evening Slot If the tangent to the parabola y2 = x at a point ($$\alpha $$,$$\beta $$), ($$\beta $$> 0) is also a tangent to the ellipse, x2 + 2y2 = 1, then ... JEE Main 2019 (Online) 8th April Evening Slot In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at (0,5$$\sqrt 3... 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 : ... 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... 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 ... 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 inte... 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 : 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 $${3 \over 2}$$... 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 : 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 bet... 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... 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, t... 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 $${{{x^2}} \over 9} + {{{y^2}... JEE Main 2014 (Offline) The locus of the foot of perpendicular drawn from the centre of the ellipse$${x^2} + 3{y^2} = 6$$on any tangent to it is : JEE Main 2013 (Offline) The equation of the circle passing through the foci of the ellipse$${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$, and having centre at$$(0,3)$$is... AIEEE 2012 STATEMENT-1 : An equation of a common tangent to the parabola$${y^2} = 16\sqrt 3 x$$and the ellipse$$2{x^2} + {y^2} = 4$$is$$y = 2x + 2\sqrt 3 $$... AIEEE 2012 An ellipse is drawn by taking a diameter of thec circle$${\left( {x - 1} \right)^2} + {y^2} = 1$$as its semi-minor axis and a diameter of the circle... AIEEE 2011 Equation of the ellipse whose axes of coordinates and which passes through the point$$(-3,1)$$and has eccentricity$$\sqrt {{2 \over 5}} $$is : AIEEE 2009 The ellipse$${x^2} + 4{y^2} = 4is inscribed in a rectangle aligned with the coordinate axex, which in turn is inscribed in another ellipse that pa... AIEEE 2008 A focus of an ellipse is at the origin. The directrix is the linex=4$$and the eccentricity is$${{1 \over 2}}$$. Then the length of the semi-major... AIEEE 2006 In the ellipse, the distance between its foci is$$6$$and minor axis is$$8$$. Then its eccentricity is : AIEEE 2005 An ellipse has$$OB$$as semi minor axis,$$F$$and$$F$$' its focii and theangle$$FBF$$' is a right angle. Then the eccentricity of the ellipse is : AIEEE 2004 The eccentricity of an ellipse, with its centre at the origin, is$${1 \over 2}$$. If one of the directrices is$$x=4$$, then the equation of the elli... ## Numerical JEE Main 2023 (Online) 15th April Morning Shift Let an ellipse with centre (1,0) and latus rectum of length \frac{1}{2} have its major axis along \mathrm{x}-axis. If its minor axis subtends an... JEE Main 2023 (Online) 1st February Evening Shift The line$$x=8$$is the directrix of the ellipse$$\mathrm{E}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$with the corresponding focus$$(2,0)$$. If t... JEE Main 2023 (Online) 24th January Morning Shift Let C be the largest circle centred at (2, 0) and inscribed in the ellipse$${{{x^2}} \over {36}} + {{{y^2}} \over {16}} = 1$$. If (1,$$\alpha$$) lie... JEE Main 2023 (Online) 24th January Morning Shift Let a tangent to the curve$$9{x^2} + 16{y^2} = 144$$intersect the coordinate axes at the points A and B. Then, the minimum length of the line segmen... JEE Main 2022 (Online) 28th July Evening Shift Let the tangents at the points$$\mathrm{P}$$and$$\mathrm{Q}$$on the ellipse$$\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$$meet at the point$$R(\sqrt{2}, ... 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 $$\lambda+... JEE Main 2022 (Online) 24th June Morning Shift If two tangents drawn from a point ($$\alpha$$,$$\beta$$) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one ... JEE Main 2021 (Online) 27th August Morning Shift If the minimum area of the triangle formed by a tangent to the ellipse$${{{x^2}} \over {{b^2}}} + {{{y^2}} \over {4{a^2}}} = 1$$and the co-ordinate ... JEE Main 2021 (Online) 27th July Evening Shift Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3,$$-$$4), one focus at (4,$$-\$4) and one vertex at (5,...
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 __________....
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