Analytic Geometry Answers

Any curve in 2D that is symmetric about the origin is also symmetric with respect to both the axes
Is the statement true? Give reason for your answers either with a
short proof or a counterexample.

Every cylinder has a circular base.
Is the statement true? Give reasons for your answers, either with a
short proof or a counterexample.

Find the locus of the point, the absolute value of the difference of the distances of
which from the points (2,2) and (0,0) is 2 . Identify the curve represented by the
locus.

Prove that the product of the distance from any point on a hyperbola to its
asymptotes is a constant.

Obtain the equation of the parabola with focus (3,2)and directrix
3x − 4y + 9 = 0 .

Under what conditions on α do the spheres x^2+y^2+z^2+αx-y=0 and x^2+y^2+z^2+x+2z+1=0 intersect each other at an angle of 45° ?

Find the equations of those tangent planes to the sphere x^2+y^2+z^2+2x-4y+6z-7=0which intersect in the line
6x-3y-23=0, 3z+2=0

If x/1=y/1=z/-1 represents one of the three mutually perpendicular generators of the cone 3xy+8xz-5yz=0, find the equations of the other two.

Prove that the paraboloids
x^2/a1^2 + y^2/b1^2 = 2z/c1
x^2/a2^2 + y^2/b2^2 = 2z/c2
x^2/a3^2 + y^2/c3^2 = 2z/c3 have a common teangent plane if
|a1^2 a2^2 a3^2|
|b1^2 b2^2 b3^2|
|c1^2 c2^2 c3^2|

Trace the surface
x^2/4 + y^2/9 - z^2/4=1
Also describe its sections by the planes x = ±2 , algebraically and geometrically.