Ellipse PYQs - Last 10 Years

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