Conic Sections ...

The equation of the circle which bisects the circumference of the circles  $$x^{2}+y^{2} =1, \;x^{2}+y^{2}-2x =3$$ and $$x^2+y^{2}+2y =3$$ is

The locus of point of intersection $$P$$ of tangents to ellipse $$2x^{2}+3y^{2}=6$$ at $$A$$ and $$B$$ if $$AB$$ subtend $$90^{o}$$ angle at centre of ellipse is an ellipse whose eccentricity is equal to 

The equation of the circle, passing through the point $$\left(2,8\right)$$, touching the lines $$4x-3y-24=0$$ and $$4x+3y-42=0$$ and having x coordinate of the centre of the circle numerically less then or equal to 8 is

A circle has ccentre $$C$$ on axes of parabola and it touches the parabola at point $$P$$. $$CP$$ makes an angle of $$120^{o}$$ with axis of parabola. If radius of circle is $$2$$, then latus rectum of parabola is 

If $${ e }_{ 1 }$$ is the eccentricity of the ellipse $$\dfrac { { x }^{ 2 } }{ 16 } +\dfrac { { y }^{ 2 } }{ 25 } =1and{ \quad e }_{ 2 }$$ is the eccentricity of the hyperbola passing through the foci of the ellipse and $${ e }_{ 1 }{ e }_{ 2 }=1$, then equation of the hyperbola is  ___________________.

Length of latus rectum of the parabola $$9x^{2}+16y^{2}+24xy-4x+3y=0$$ is :

Find the equation of the circle having $$(1, -2)$$ as its centre and passing through the intersection of the lines $$3x+y=14$$ and $$2x+5y=18$$.

The equation of the circle passing through $$(4,6)$$ and having centre $$(1,2)$$ is

If the latus rectum subtends a right angel at the center of a hyperbola, then its eccentricity is 

If the length of the latus rectum of the parabola $$289(x-3)^{2}+289(y-1)^{2}=(15x-8y+13)^{2}$$ is $$k$$, then $$[k]=?$$