Conic Sections ...

The ends of the latus rectum of the parabola $$x^{2} + 10x - 16y + 25 = 0$$ are

If $$e_{1}$$ and $$e_{2}$$ are the eccentricities of two conics with $$e_{1}^{2} + e_{2}^{2} = 3$$, then the conics are.

The line segment joining the foci of the hyperbola $$x^{2} - y^{2} + 1 = 0$$ is one of the diameters of a circle. The equation of the circle is :

Consider the conic $$e{ x }^{ 2 }+\pi { y }^{ 2 }-2{ e }^{ 2 }x-2{ \pi  }^{ 2 }y+{ e }^{ 3 }+{ \pi  }^{ 3 }=\pi e$$. Suppose $$P$$ is any point on the conic and $${ S }_{ 1 }, { S }_{ 2 }$$ are the foci of the conic, then the maximum value of $$\left( P{ S }_{ 1 }+P{ S }_{ 2 } \right) $$ is

The eccentricity of an ellipse $$9{ x }^{ 2 }+16{ y }^{ 2 }=144$$ is

The directrix of a parabola is $$x+8=0$$ and its focus is at $$(4,3)$$. Then, the length of the latusrectum of the parabola is

The one end of the latusrectum of the parabola $${ y }^{ 2 }-4x-2y-3=0$$ is at

The foci of the ellipse $$4{x}^{2}+9{y}^{2}=1$$ are

If the eccentricity of the hyperbola $${ x }^{ 2 }-{ y }^{ 2 }\sec ^{ 2 }{ \alpha  } =5$$ is $$\sqrt { 3 } $$ times the eccentricity of the ellipse $${ x }^{ 2 }\sec ^{ 2 }{ \alpha  } +{ y }^{ 2 }=25$$, then the value of $$\alpha $$ is

If the eccentricity of the ellipse $$a{ x }^{ 2 }+4{ y }^{ 2 }=4a,(a<4)$$ is $$\cfrac { 1 }{ \sqrt { 2 }  } $$, then its semi-minor axis is equal to