Ellipse - Coordinate Geometry PYQs (Math)

In an ellipse, if the distance between the foci is 6 units and the length of its minor axis is 8 units, then its eccentricity is

If a point \(P(x, y)\) moves along the ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\) and if \(C\) is the center of the ellipse, then the sum of maximum and minimum values of \(C P\) is

A point moves so that the sum of its distances from \((a e, 0)\) and \((-a e, 0)\) is \(2 a\), then the equation to its locus, where \(b^2=a^2\left(1-e^2\right)\) is

If \(\tan \theta_1, \tan \theta_2=\frac{-a^2}{b^2}\), then the chord joining 2 points \(\theta_1\) and \(\theta_2\) one the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) will subtend a right angle at

The focal distances of the point \(\left(\frac{4}{\sqrt{5}}, \frac{3}{\sqrt{5}}\right)\) on the ellipse \(\frac{x^2}{4}+\frac{y^2}{9}=1\) are

A stick of length \(r\) units slides with its ends on coordinate axes. Then, the locus of the mid-point of the stick is a curve whose length is

The eccentric angle of a point on the ellipse \(x^2+3 y^2=6\) lying at a distance of 2 units from its centre is

If the angle between the straight lines joining the foci and the ends of the minor axis of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is \(90^{\circ}\), then it eccentricity

If the tangents are drawn to the ellipse $x^2+2 y^2=2$, then the locus of the mid-points of the intercepts made by the tangents between the coordinate axes is

The equation of a chord $A B$ of an ellipse $2 x^2+y^2=1$ is $x-y+1=0$. If $O$ is the origin, then $\sqrt{A O B}=$

The square of the slope of a common tangent drawn to the circle $4 x^2+4 y^2=25$ and the ellipse $4 x^2+9 y^2=36$ is

Let $A_1$ be the area of the given ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Let $A_2$ be the area of the region bounded by the curve which is the locus of mid-point of the line segment joining the focus of the ellipse and a point $P$ on the given ellipse, then $A_1: A_2=$