Show that the equation `y^2 + 6y - 2x + 5 = 0` represents parabola. Find its vertex, focus, length of latus-rectum, equations of axis and directrix.

Show that the equation y^2 - 8y -x+19 = 0 represents a parabola. Find its vertex, focus and directrix.

The equation ax^2+4xy+y^2+ax+3y+2=0 represents a parabola. Find the value of a .

Prove that the equation y^2+ 2Ax + 2By +c =0 represents a parabola whose axis is parallel to the x-axis.

Show that the equation : 16x^2 -3y^2-32x -12y- 44 = 0 represents a hyperbola , and find the lengths of the axes and eccentricity.

Find the vertex, focus, latus-rectum, axis and directrix of the parabola x^2- y - 2x = 0 .

Find the vertex and length of latus rectum of the parabola x^(2)=-4(y-a) .

Prove that the equation y^(2)+2ax+2by+c=0 represents a parabola whose axis is parallel to the axis of x. Find its vertex.

Find the length of latus-rectum of the following condition : Focus (6,0), Directrix x =- 6.

Show that 4x^2+8x+y^2-4y+4=0 represents an ellipse. Find its eccentricity, co-ordinates of foci, equations of major and minor axes and latus-rectum.

Show that the differntial equation that represents all parabolas each of which has a latus rectum 4a and whose axes are parallel to x axis is 2ay_2+y_1^3=0 .