Ellipse — JEE Main PYQs

Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along $x$-axis. If its minor axis subtends an angle $60^{\circ }$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to

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

Consider ellipses $E_k:k{x}^2+k^2{y}^2=1,k=1,2,\dots ,20$. Let $C_k$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_k$. If $r_k$ is the radius of the circle $C_k$, then the …

Let a tangent to the Curve $9x^2+16y^2=144$ intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is

The line $\mathrm{x}=8$ is the directrix of the ellipse $\mathrm{E}:\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ with the corresponding focus $(2,0)$. If the tangent to E at the point P in the first quadrant passes through the point …

If the tangent at a point P on the parabola $y^2=3x$ is parallel to the line $x+2y=1$ and the tangents at the points Q and R on the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ are perpendicular to the line $x-y=2$, then the area of the triangle PQR is:

If the maximum distance of normal to the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1,b<2$, from the origin is 1, then the eccentricity of the ellipse is:

Let the tangents at the points P and Q on the ellipse $\frac{{\mathrm{x}}^2}{2}+\frac{{\mathrm{y}}^2}{4}=1$ meet at the point $\mathrm{R}(\sqrt{2},2\sqrt{2}-2)$. If S is the focus of the ellipse on its negative major axis, then ${\mathrm{SP}}^2+{\mathrm{SQ}}^2$ is equal t …

An ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $H:\frac{x^2}{49}-\frac{y^2}{64}=-1$. Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H. Let the product of t …

If the length of the latus rectum of the ellipse $x^2+4y^2+2x+8y-\lambda =0$ is 4, and $l$ is the length of its major axis, then $\lambda +l$ is equal to ______.

The acute angle between the pair of tangents drawn to the ellipse $2x^2+3y^2=5$ from the point $(1,3)$ is

A common tangent $T$ to the curves $C_1:\frac{x^2}{4}+\frac{y^2}{9}=1$ and $C_2:\frac{x^2}{42}-\frac{y^2}{143}=1$ does not pass through the fourth quadrant. If $T$ touches $C_1$ at $(x_1,y_1)$ and $C_2$ at $(x_2,y_2)$, then $|2x_1+x_2|$ is equal to ____.

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, then the eccentricity of the ellipse is

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 $12m^2$ is equal to :

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

Let the common tangents to the curves $4(x^2+y^2)=9$ and $y^2=4x$ intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and $l$ respectively denote the eccentricity and t …

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 $\left(-4\sqrt{\frac{2}{5}},3\right)$, then ${\mathrm{a}}^2+{\mathrm{b}}^2$ is equal to :

Let a line L pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0$, $\mathrm{b}\in \mathrm{R}-\left\{\frac{4}{3}\right\}$. If the line L also passes through the point $(1,1)$ and touches the circle $17\left(x^2+y^2\right)=16$, then the ecce …

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 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 eccentricit …

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 circle with PQ as diameter then $(3r{)}^2$ is equal to

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 the ellipse is perpendicular to the line $x+2y=0$. Let $S$ and $S^{\prime }$ be the foci of the ellipse and e be its eccentricity. If $A$ is the area of th …

Let $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$. Let $E_2$ be another ellipse such that it touches the end points of major axis of $E_1$ and the foci of $E_2$ are the end points of minor axis of $E_1$. If $E_1$ and $E_2$ have same eccentricities, then its value is:

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

If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^2}{b^2}+\frac{y^2}{4a^2}=1$ and the coordinate axis is kab , then k is equal to

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

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:

The line $12x\cos \theta +5y\sin \theta =60$ is tangent to which of the following curves?

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 any ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is …

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 $\frac{2}{\sqrt{3}}$, …

If a tangent to the ellipse $x^2+4y^2=4$ meets the tangents at the extremities of its major axis at $B$ and $C$, then the circle with $BC$ as diameter passes through the point :

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 $5{\mathrm{m}}^2$ is equal to ______

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 of intersection in the first quadrant. The $\tan \theta$ is equal to:

If the points intersections of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the circle $x^2+y^2=4b,b>4$ lie on the curve $y^2=3x^2$, then $b$ is equal to :

Let a tangent be drawn to the ellipse $\frac{x^2}{27}+y^2=1$ at $(3\sqrt{3}\cos \theta ,\sin \theta )$ where $\theta \in \left(0,\frac{\pi }{2}\right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

Let $L$ be a tangent line to parabola $y^2=4x-20$ at $(6,2)$. If $L$ is also a tangent to the ellipse $\frac{x^2}{2}+\frac{y^2}{b}=1$, then the value of $b$ is equal to:

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:

If $3x+4y=12\sqrt{2}$ is a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{9}=1$ for some $a\in R$, then the distance between the foci of the ellipse is:

Let the line $\mathrm{y}=\mathrm{mx}$ and the ellipse $2{\mathrm{x}}^2+{\mathrm{y}}^2=1$ intersect at a point $P$ in the first quadrant. If the normal to this ellipse at P meets the co - ordinate axes at $\left(-\frac{1}{3\sqrt{2}},0\right)$ and $(0,\beta )$ then $\beta$ is eq …

The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line, $x+6y=8$; then its eccentricity is:

If $e_1$ and $e_2$ are the eccentricities of the ellipse, $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola, $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and $(e_1,e_2)$ is a point on the ellipse, $15x^2+3y^2=k$, then $k$ is equal to:

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:

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:

If the point $P$ on the curve, $4x^2+5y^2=20$ is farthest from the point $Q(0,-4)$ then $P{Q}^2$ is equal to:

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac{x^2}{4}+\frac{y^2}{2}=1$ from any of its foci?

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ be given ellipse, length of whose latus rectum is 10 . If its eccentricity is the maximum value of the function, $\varphi (t)=\frac{5}{12}+t-t^2$, then $a^2+b^2$ is equal to:

Let $\mathrm{x}=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac{1}{2}$. If $\mathrm{P}(1,\beta ),\beta >0$ is a point on this ellipse, then the equation of the normal to it at P is:

Let $\mathrm{O}(0,0)$ and $\mathrm{A}(0,1)$ be two fixed points. Then the locus of a point P such that the perimeter of $△AOP$, is 4, is:

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

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:

Let the length of the latus rectum of an ellipse with its major axis along x-axis and center 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?