Analytic Geometry Answers

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

find the equation of the normal to the parabola y^2+4x=0 at the point where the line y=x+c touches it

Find the equation of the tangent plane to the conicoid x^2+y^2 = kz at the point
(k;k;2k), where k is a constant. Represent the plane geometrically. Now take
different values of k, including both positive and negative, and see how the shape of
the conicoid changes.

Identify and trace the conicoid y^2+z^2 = x. Describe its sections by the planes
x = 0;y = 0 and z = 0.

Find the vertices, eccentricity, foci and asymptotes of the hyperbola x2
8

Equation of one side of a square is 2x+3y+4=0. If the center is (1,1) then find the equations of adjoined two sides of the square.

In a square ABCD, A(1,3) and C(4,2). AC is a diagonal. Express the coordinates of a point on the diagonal BD using a real parameter. Hence find the coordinates of the other two vertices.

(a^2+b^2)\ab+1=4
Give a equation to solve the value of a and b.
a=? & b?