Conic Sections ...

The hyperbola $$\cfrac { { x }^{ 2 } }{ { a }^{ 2 } } -\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$$ passes through the point $$\left( \sqrt { 6 } ,3 \right) $$ and the length of the latusrectum is $$\cfrac { 18 }{ 5 } $$. Then, the length of the transverse axis is equal to

The radius of the circle passing through the points $$(2,3),(2,7)$$ and $$(5,3)$$ is

The equation of the circumcircle of the triangle formed by the lines $$y + \sqrt{3} x = 6, y - \sqrt{3} x = 6$$ and $$y=0$$ is 

The equation of the circle having $$x-y-2=0$$ and $$x-y+2=0$$ as two tangents and $$x-y=0$$ as diameter is

The lines $$y=x+\sqrt { 2 } $$ and $$y=x-2\sqrt { 2 } $$ are the tangent of certain circle. If the point $$\left( 0,\sqrt { 2 }  \right) $$ lies on this circle, then its equation is

On the parabola $$y={ x }^{ 2 }$$, the point least distant from the straight line $$y=2x-4$$ is

The curve represented by $$x = 3\cos t + 3\sin t$$ and $$y = 4\cos t - 4\sin t$$ is

The lines $$2x-3y=5$$ and $$3x-4y=7$$ are the diameters of a circle of area $$154$$ sq.units. The equation of the circle is

The equation of the circle which touches the lines $$x = 0, y = 0$$ and $$4x + 3y = 12$$ is

Equations $$x = a\cos \theta$$ and $$y= b\sin \theta$$ represent a conic section whose eccentricity $$e$$ is given by