Ellipse Questions (292)

If \(\beta\) is one of the angles between the normal to the ellipse, \(x^2 + 3y^2 = 9\) at the points \((3\cos\theta, \sqrt{3}\sin\theta)\) and \((-3\sin\theta, \sqrt{3}\cos\theta)\); \(\theta \in (0, \pi/2)\); then \(\dfrac{2\cot\beta}{\sin 2\theta}\) is equal to

Show that the equation of the locus of a point which moves so that the sum of its distances from two given points \((ae, 0)\) and \((-ae, 0)\) is equal to \(2a\), is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) where \(b^2 = a^2(1 - e^2)\).

The equation \(14x^2 - 4xy + 11y^2 - 44x - 58y + 71 = 0\) represents

If the tangent to the ellipse \(x^2 + 4y^2 = 16\) at the point \(P\phi\) is a normal to the circle \(x^2 + y^2 - 8x - 4y = 0\), then \(\phi\) is equal to

If the area of the quadrilateral formed by the tangents at the ends of the latus rectum of the ellipse \(E\) is \(\dfrac{16\lambda}{\sqrt{55}}\), then \(\lambda\) equals:

The foci of the ellipse \(\dfrac{x^2}{16} + \dfrac{y^2}{b^2} = 1\) and the hyperbola \(\dfrac{x^2}{144} - \dfrac{y^2}{81} = \dfrac{1}{25}\) coincide. Then the value of \(b^2\) is

The angle between the pair of tangents drawn to the ellipse $3x^2 + 2y^2 = 5$ from the point $(1, 2)$ is $\tan^{-1}\left(\frac{4}{5}\right)$, then the value of $\lambda$ is

If the line $x - 2y = 12$ is a tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $\left(3, -\frac{9}{2}\right)$, then the length of the latus rectum of the ellipse is

If congruent tangents of $x^2 + y^2 = r^2$ and $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ form a square, then the length of diagonal of the square is

If the value of \(\displaystyle\sum_{i=1}^{n} \frac{\text{Area}(\Delta P_i T_i S) \cdot \text{Area}(\Delta P_i T_i S')}{(P_i T_i)^2} = 18\), where \(S\) and \(S'\) represents the foci of the ellipse, then \(n\) equal to:

In the above problem if \(m = \frac{\text{area of } \triangle PQR}{\text{area of } \triangle PQS}\), then \(2m\) =

If the line \(2px + y\sqrt{1-p^2} = 1\) always touches the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) \(\forall\, p \in (-1,1) - \{0\}\). The eccentricity of this ellipse, is

If \frac{x}{a cos θ} + \frac{y}{b sin θ} = 1 and \frac{x}{a sin θ} - \frac{y}{b cos θ} = a² - b², then (x, y) lie on

If the line x - 2y = 12 is tangent to the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) at the point \(\left(3, \frac{-9}{2}\right)\), then the length of the latusrectum of the ellipse is

A normal is drawn to the ellipse \(\frac{x^2}{(a^2 + 2a + 2)^2} + \frac{y^2}{(a^2 + 1)^2} = 1\) whose centre is at O. If maximum radius of the circle, centered at the origin and touching the normal, is 5 then find the positive value of 'a'.

The maximum number of normals that can be drawn to an ellipse/hyperbola passing through a given point is:

If a rectangular hyperbola \((x-1)(y-2)=4\) cuts a circle \(x^2+y^2+2 g x+2 f y+c=0\) at points \((3,4),(5,3),(2,6)\) and \((-1,0)\), then the value of \((g+f)\) is equal to

Let O(0, 0) and A(0, 1) be two fixed points, then the locus of a point P such that the perimeter of △AOP is 4, is

Point \(O\) is the centre of the ellipse with major axis \(AB\) and minor axis \(CD\). Point \(F\) is one focus of the ellipse. If \(OF = 6\) and the diameter of the inscribed circle of triangle \(OCF\) is \(2\), then the product \((AB)(CD)\) is equal to:

The eccentricity of an ellipse whose centre is at the origin is 1/2. If one of its directrices is x = -4, then the equation of the normal to it at \(\left(1, \frac{3}{2}\right)\) is

It is given that e1 is the eccentricity of the ellipse x2 + y2/4 = 1, and e2 is the eccentricity of the hyperbola x2/9 - y2/4 = 1. If the point (e1, e2) lies on the curve 15x2 + 3y2 = k, find the value of k.

The locus of the middle points of chords of an ellipse , which passes through a fixed point, is

The area (in sq units) of the quadrilateral formed by the tangents at the end points of the latusrectum to the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{5}=1\) is

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

An ellipse and a hyperbola have the same foci. If e1 is the eccentricity of the ellipse and e2 is the eccentricity of the hyperbola, and D = e2 − e1, then for \(\frac{1}{2} D is

If a tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), whose centre is C meets the major and minor axes produced at P and Q respectively then \(\frac{a^2}{CP^2} + \frac{b^2}{CQ^2}\) is equal to ___.

For the ellipse 4(x - 2y + 1)2 + 9(2x + y + 2)2 = 180, lengths of major and minor axes are respectively

The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse \(x^2 + 9y^2 = 9\) meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin O is:

Let P be a variable point on the ellipse \(\frac{x^2}{25} + \frac{y^2}{4} = 2\) with foci \(F_1\) and \(F_2\). If A is the area of the \(\triangle PF_1F_2\) then the maximum value of A is ___.

For the ellipse \(\frac{x^2}{a_n^2} + \frac{y^2}{b_n^2} = 1\), the tangent at point \((x_1, y_1)\) is \(T = S_1\), i.e., \(\frac{xx_1}{a_n^2} + \frac{yy_1}{b_n^2} = \frac{x_1^2}{a_n^2} + \frac{y_1^2}{b_n^2}\). Given that \(b_n^2 x_1 = a_n^2 y_1\) and the eccentricity \(e = \frac{\sqrt{5}-1}{2}\), find the relation between \(x_1\) and \(y_1\).

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

The sum of the squares of the perpendiculars on any tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) from two points on the minor axis each at a distance \(a^2 - b^2\) from the centre is

The line $2x + y = 3$ cuts the ellipse $4x^2 + y^2 = 5$ at points $P$ and $Q$. If $\theta$ is the acute angle between the normals at $P$ and $Q$, then $\theta$ 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 __________.

If the eccentricity of an ellipse be \(\frac{1}{\sqrt{2}}\), then its latus rectum is equal to its

If \(y = x\) and \(3y + 2x = 0\) are the equations of a pair of conjugate diameters of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) then the eccentricity is ___.

If the normal to the ellipse \(3x^2 + 4y^2 = 12\) 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 __________ (up to two decimal places).

If the middle point of a chord of the ellipse \(\frac{x^2}{25} + \frac{y^2}{16} - 1\) be \(\left[\frac{2}{5}, \frac{4}{5}\right]\) then the length of the chord will be ___.

Let the eccentricity of an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ be reciprocal to that of the hyperbola $2x^2-2y^2=1$. If the ellipse intersects the hyperbola at right angles, then the square of the length of the latus-rectum of the ellipse is _____.

Let the ellipse $E:\ x^2+9y^2=9$ intersect the positive $x$- and $y$-axes at $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line through $A$ and $B$ meet $C$ at $P$. If the area of $\triangle OAP$ is $\dfrac{m}{n}$ where $m,n$ are coprime, then $m-n$ is equal to

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

Let the product of the focal distances of the point (\sqrt3, 2 y on the ellipse , be . Then the 1 x 7 ) + = 1, (a > b) 2 a 2 b 2 4 absolute difference of the eccentricities of two such ellipses is 1-\sqrt3

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

If the points of intersection of two distinct conics $x^2+y^2=4b$ and $\dfrac{x^2}{16}+\dfrac{y^2}{b^2}=1$ lie on the curve $y^2=3x^2$, then $3\sqrt{3}$ times the area of the rectangle formed by the intersection points is

Let \(P Q\) be a focal chord of the parabola \(y^{2}=4 x\) such that it subtends an angle of \(\frac{\pi}{2}\) at the point \((3,0)\). Let the line segment \(P Q\) be also a focal chord of the ellipse \({E}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2} \gt b^{2}\). If \(e\) is the eccentricity of the ellipse \(E\), then the value of \(\frac{1}{e^{2}}\) is equal to:

The equation of the chord, of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$, whose mid-point is $(3, 1)$ is

If a focus = (6, 7), the corresponding directrix is \(x + y + 2 = 0\) and eccentricity = \(\frac{1}{\sqrt{3}}\) then the equation of the ellipse is ___.

Let S(5,12) and S'(−12,5) are the foci of an ellipse passing through the origin. The eccentricity of ellipse equals -

If \(\left(a, \frac{1}{a}\right)\) and \(\left(b, \frac{1}{b}\right)\) be the extremities of chord on hyperbola \(x y=1\), then line perpendicular to this chord will be equally inclined to co-ordinate axes if

Consider the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$. Let $S(p,q)$ be in the first quadrant with $\dfrac{p^2}{9}+\dfrac{q^2}{4}>1$. Two tangents from $S$: one at a minor axis endpoint, the other at $T$ in the fourth quadrant. $R$ = vertex with positive $x$-coordinate, $O$ = center. If area of $\triangle ORT=\dfrac{3}{2}$, then: