Ellipse practice · Easy

For the ellipse $25 x^2+9 y^2-150 x-90 y+225=0$, the eccentricity is

Locus of the point of intersection of tangents at the end points of focal chord is

If $x=6 y+8$ be a tangent to the ellipse in standard form whose minor axis is $\frac{4}{\sqrt{3}}$, then its eccentricity is

The eccentricity of the ellipse represented by the equation $25\;{x^2} + 16\;{y^2} - 150\;x - 175 = 0$ is

The eccentricity of the ellipse $4 x^2+y^2-8 x+2 y+1=0$ is

The equation of the ellipse, whose one of the vertices is $(0,7)$ and the corresponding directrix is $y=12$, is

Let a line $L$ pass through the point intersection of the lines $b x+10 y-8=0$ and $2 x-3 y=0$, $b \in R-\left\{\frac{4}{3}\right\}$. If the line $L$ also passes through the point $(1,1)$ and touches the circle $17\left(x^2+y^2\right)=16$, then the eccentricity of the ellipse $\frac{x^2}{5}+\frac{y^2}{5}=1$ is

If $P$ is any point on the ellipse $\frac{{{x^2}}}{{36}} + \frac{{{y^2}}}{{16}} = 1$ and $S$ and $S'$ are the foci, then $PS + PS'$ is equal to

If polar of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is always touching the circle $x^2+y^2=c^2$, then locus of pole is

Coordinates of the foci of the ellipse $5{x^2} + 9{y^2} + 10x - 36y - 4 = 0$ are

A tangent to the ellipse $4 x^2+9 y^2=36$ is cut by the tangent at the extremities of the major axis at $T$ and ' $T$ '. The circle on $T T^{\prime}$ as diameter passes through the point

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is:

The line $l x+m y-n=0$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ if

The locus of the point which moves such that the ratio of its distance from two fixed point in the plane is always a constant $k( < 1)$ is

If $P=(x, y), F_1=(3,0), F_2=(-3,0)$ and $16 x^2+25 y^2=400$, then $P F_1+P F_1$ equals:

The equation of the ellipse (referred to its axes as the axes of $x$ and $y$ respectively) which passes through the point $\left( { - 3,1} \right)$ and has eccentricity $\sqrt {\frac{2}{5}} ,$ is

The line $l x+m y+n=0$ is a normal to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, if

The length of latusrectum of the ellipse $9{x^2} + 16{y^2} = 144$ is

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

If the eccentricities of the ellipse $\frac{{{x^2}}}{4} + \frac{{{y^2}}}{3} = 1$ and the hyperbola $\frac{{{x^2}}}{{64}} - \frac{{{y^2}}}{{{b^2}}} = 1$ are reciprocals of each other, then ${b^2}$ is equal to

The eccentricity of the ellipse $25 x^2+16 y^2-150 x-175=0$ is

The dimensions of the rectangle of maximum area that can be inscribed in the ellipse $(x / 4)^2+(y / 3)^2$ $=1$ are

The equation of the ellipse whose distance between foci is equal to 8 and distance between the directrix is 18, is

Equation of a line joining a foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ to a foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ is,

If $y=x+1$ is a polar to $x^2+2 y^2=1$ then pole is

The locus of the point of intersection of perpendicular tangents to $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1$, is

An ellipse is drawn by taking a diameter of the circle $(x-1)^2+y^2=1$ as its semi-minor axis and diameter of the circle $x^2+(y-2)^2=4$ is semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is

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

If the length of the major axis of an ellipse is three times the length of its minor axis, its eccentricity, is

The eccentricity of the ellipse $9 x^2+5 y^2-30 y=0$ is