Analytic Geometry Answers

The equations 2x^2+y^2+3z^2+4x+4y+18z+34=0, 2x^2-y^2=4y-4z-4x represent a real conic.
Whether the statement is true or false.
Give reasons for your answer.

Which of the following statements are true and which are false? Give reasons for your
answer.
i) The equation r = acos(theta+alpha)+bsin(theta +alpha) represents a circle.
ii) The direction ratios of the line x-5 = 5-y; z = 5 are 1;1;5:
iii) The intersection of a plane and a cone can be a pair of lines.

1.Find the equation of the normal to the parabola y^2+4x=0 at the point where the line y=x+c touches it. Q.(2) Identify and trace the conic x^2-2xy+y^2-3x+2y+3=0.

Identify and trace the conicoid y^2+z^2=x.
Describe it's sections by the planes x=0,y=0 and z=0.

Find the transformation of the equation 12x^2-2y^2+z^2=2xy if the origin is kept fixed and the axes are rotated in such a way that the direction ratios of the new axes are 1,-3,0; 3,1,0; 0,0,1.

Find the equations of the spheres which pass through the circle x^2+y^2+z^2=9, 2x+2y-7=0 and the touch the plane x-y+z+3=0

Obtain the equation of the conic, a focus of which lies at (2,1), the directrix of which is x+y=0 and which passes through (1,4). Also identify the conic.

Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.
1) x^2+y^2+z^2+x+y+z=1
2) 2x^2+4xy+xz-x-3y+5z+3=0
3) x^2-y^2-z^2+xy+4yz+x=0

Prove that the plane 2x-3y+6z=6 touches the conicoid 4x^2-9y^2+36z^2=36.
Find the point of contact.

identify and trace the conic x2-2xy+y2-3x+2y+3=0