Ellipse - Coordinate Geometry - Mathematics Previous Year Questions

A line segment joining a point $A$ on $X$-axis to a point $B$ on $Y$-axis is such that $A B=15$. If $P$ is a point on $A B$ such that $\frac{A P}{P B}=\frac{2}{3}$, then the locus of $P$ is

If any tangent drawn to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ touches one of the circles $x^2+y^2=\alpha^2$, then the range of $\alpha$ is

One of the foci of an ellipse is $(2,-3)$ and its corresponding directrix is $2 x+y=5$. If the eccentricity of the ellipse is $\frac{\sqrt{5}}{3}$, then the coordinates of the other focus are

If $S$ and $S^{\prime}$ are the foci of an ellipse $\frac{x^2}{169}+\frac{y^2}{144}=1$ and the point $B$ lying on positive $Y$-axis is one end of its minor axis, then the incentre of the $\triangle S B S^{\prime}$ is

The length of the chord of the ellipse $\frac{x^2}{4}+y^2=1$ formed on the line $y=x+1$ is

If the perpendicular distance from the focus of an ellipse $\frac{x^2}{9}+\frac{y^2}{b^2}=1(b<3)$ to its corresponding directrix is $\frac{4}{\sqrt{5}}$, then the slope of the tangent to this ellipse drawn at $\left(\frac{3}{\sqrt{2}}, \frac{b}{\sqrt{2}}\right)$ is

Let $P$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and let the perpendicular drawn through $P$ to the major axis meet its auxiliary circle at $Q$. If the normals drawn at $P$ and $Q$ to the ellipse and the auxiliary circle respectively meet in $R$, then the equation of the locus of $R$ is

The mid-point of the chord of the ellipse $x^2+\frac{y^2}{4}=1$ formed on the line $y=x+1$ is

If a normal is drawn at a variable point $P(x, y)$ on the curve $9 x^2+16 y^2-144=0$, then the maximum distance from the centre of the curve to the normal is

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(b>a)$ is an ellipse with eccentricity $\frac{1}{\sqrt{2}}$. If the angle of intersection between the ellipse and parabola $y^2=4 a x$ is $\theta$, then the coordinates of the point $\frac{2 \theta}{3}$ on the ellipse is