For the ellipse `4x^(2)+y^(2)+24x-2y+21=0`, find the centre, vertices, and the foci. Also prove that the length of latus rectum is 2.

For the ellipse 4x^(2)+3y^(2)+8x+12y+4=0 . Find the centre, vertices, foci and latus rectum.

Find the equations of normal to the parabola y^2=4a x at the ends of the latus rectum.

Find the value of lambda if the equation (x-1)^2+(y-2)^2=lambda(x+y+3)^2 represents a parabola. Also, find its focus, the equation of its directrix, the equation of its axis, the coordinates of its vertex, the equation of its latus rectum, the length of the latus rectum, and the extremities of the latus rectum.

If the line y = 3x + 1, touches the parabola y^(2) = 4ax , find the length of the latus rectum ?

The value of a for the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1,(a > b), if the extremities of the latus rectum of the ellipse having positive ordinates lie on the parabola x^2=2(y-2) is ___

y^2+2y-x+5=0 represents a parabola. Find its vertex, equation of axis, equation of latus rectum, coordinates of the focus, equation of the directrix, extremities of the latus rectum, and the length of the latus rectum.

If the ellipse x^(2)+2y^(2)=4 and the hyperbola S = 0 have same end points of the latus rectum, then the eccentricity of the hyperbola can be

Find the area of the parabola y^2=4ax and its latus rectum.

If the normal at one end of the latus rectum of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 passes through one end of the minor axis, then prove that eccentricity is constant.