class 11

Math

Co-ordinate Geometry

Conic Sections

The two circles $x_{2}+y_{2}=ax$and $x_{2}+y_{2}=c_{2}(c>0)$touch each other if :

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If the normal at any point $P$ on the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ meets the axes at $Gandg$ respectively, then find the raio $PG:Pg=$ (a) $a:b$ (b) $a_{2}:b_{2}$ (c) $b:a$ (d) $b_{2}:a_{2}$

A tangent having slope of $−34 $ to the ellipse $18x_{2} +32y_{2} =1$ intersects the major and minor axes at points $AandB,$ respectively. If $C$ is the center of the ellipse, then find area of triangle $ABC˙$

If $C$ is the center of the ellipse $9x_{2}+16y_{2}=144$ and $S$ is a focus, then find the ratio of $CS$ to the semi-major axis.

Prove that any point on the ellipse whose foci are $(−1,0)$ and $(7,0)$ and eccentricity is $21 $ is $(3+8cosθ,43 sinθ),θ∈R˙$

Find the center, foci, the length of the axes, and the eccentricity of the ellipse $2x_{2}+3y_{2}−4x−12y+13=0$

Find the equation of the ellipse (referred to its axes as the axes of $xandy$ , respectively) whose foci are $(±2,0)$ and eccentricity is $21 $

Find the coordinates of the foci and the vertices the eccentricity and the length of the latus of rectum of the hyperbola $9y_{2} −27x_{2} =1$

$AOB$ is the positive quadrant of the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ in which $OA=a,OB=b$ . Then find the area between the arc $AB$ and the chord $AB$ of the ellipse.