Ellipse

The minimum length of intercept on any tangent to the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ cut by the circle $x^2+y^2=25$ is

Consider the following ellipse :
$\frac{x^2}{f\left(K^2+2 K+5\right)}+\frac{y^2}{f(K+11)}=1$, where $f(x)$ is a positive decreasing function. Then the value (values) of $K$ for which the major axis coincides with $x$-axis is

With origin as a focus and \(x=4\) as corresponding directrix, a family of ellipse are drawn. Then the locus of an end of minor axis is

A line of fixed length \(\mathrm{a}+\mathrm{b} . \mathrm{a} \neq \mathrm{b}\) moves so that its ends are always on two fixed perpendicular straight lines. The locus of a point which divides the line into two parts of length a and b is

The equation \(\mathrm{r} \cos \theta=2 \mathrm{a} \sin ^2 \theta\) represents the curve

Let f be a strictly decreasing function defined on R such that \(f(x) > 0,\forall x \in R\). Let \({{{x^2}} \over {f({a^2} + 5a + 3)}} + {{{y^2}} \over {f(a + 15)}} = 1\) be an ellipse with major axis along the y-axis. The value of 'a' can ...

If the lines joining the focii of the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) where \(a > b\), and an extremity of its minor axis is inclined at an angle 60\(^\circ\), then the eccentricity of the ellipse is

The tangent at point \((a\cos \theta ,b\sin \theta ),0 < \theta < {\pi \over 2}\), to the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) meets the x-axis at T and y-axis at T\(_1\). Then the value of $$\mathop {\min }\l...

Chords of an ellipse are drawn through the positive end of the minor axis. Their midpoint lies on

AB is a variable chord of the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\). If AB subtends a right angle at the origin O, then \({1 \over {O{A^2}}} + {1 \over {O{B^2}}}\) equals to

The points of intersection of two ellipses \({x^2} + 2{y^2} - 6x - 12y + 20 = 0\) and \(2{x^2} + {y^2} - 10x - 6y + 15 = 0\) lie on a circle. The centre of the circle is

The co-ordinate of a point on the auxiliary circle of the ellipse x2 + 2y2 = 4 corresponding to the point on the ellipse whose eccentric angle is 60\(^\circ\) will be

Consider the curve \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\). The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

Consider the curve \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\). The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

If B and B' are the ends of minor axis and S and S' are the foci of the ellipse \({{{x^2}} \over {25}} + {{{y^2}} \over 9} = 1\), then the area of the rhombus SBS' B' will be

Consider a tangent to the ellipse \({{{x^2}} \over 2} + {{{y^2}} \over 1} = 1\) at any point. The locus of the mid-point of the portion intercepted between the axes is

P is the extremity of the latusrectum of ellipse \(3{x^2} + 4{y^2} = 48\) in the first quadrant. The eccentric angle of P is

S and T are the foci of an ellipse and B is the end point of the minor axis. If STB is equilateral triangle, the eccentricity of the ellipse is

Let P be a point on the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\) and the line through P parallel to the Y-axis meets the circle x2 + y2 = 9 at Q, where P, Q are on the same side of the X-axis. If R is a point on PQ such that $$...

B is an extremity of the minor axis of an ellipse whose foci are S and S'. If \(\angle SBS'\) is a right angle, then the eccentricity of the ellipse is

Tangents are drawn to the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 5} = 1\) at the ends of both latusrectum. The area of the quadrilateral, so formed is

The equation of auxiliary circle of the ellipse \(16{x^2} + 25{y^2} + 32x - 100y = 284\) is

The line y = x + \(\lambda\) is tangent to the ellipse 2x2 + 3y2 = 1. Then, \(\lambda\) is

The points of the ellipse 16x2 + 9y2 = 400 at which the ordinate decreases at the same rate at which the abscissa increases is/are given by

On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y, are

The equation 8x2 + 12y2 \(-\) 4x + 4y \(-\) 1 = 0 represents

The length of the latus rectum of the ellipse is 16x2 + 25y2 = 400 is

S and T are the foci of an ellipse and B is an end point of the minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is

The angle between the line joining the foci of an ellipse to one particular extremity of the minor axis is 90\(^\circ\). The eccentricity of the ellipse is

The line y = 2t2 intersects the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\) in real point if

The total number of tangents through the point (3, 5) that can be drawn to the ellipse 3x2 + 5y2 = 32 and 25x2 + 9y2 = 450 is

The latus rectum of an ellipse is equal to one-half of its minor axis. The eccentricity of the ellipse is

The equation of the ellipse having vertices at (\(\pm\) 5, 0) and foci (\(\pm\) 4, 0) is

If 2y = x and 3y + 4x = 0 are the equations of a pair of conjugate diameters of an ellipse, then the eccentricity of the ellipse is