Analytic Geometry Answers

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.

Check whether or not the conicoid represented by
5x^2+4y^2-4yz+2xz+2x-4y-8z+2=0 is central or not. If it is, transform the
equation by shifting the origin to the centre. Else, change any one coefficient to make the equation that of a central conicoid.

Q: x = y = z −1 does not intersect the cone x^2+y^2+z^2+2(xy+yz+zx)=0
Is the statement true? Give reason for your answer, either with a short proof or a counterexample.

Q: x + y + z = 0 touches the cone x^2+y^2+z^2+2(xy+yz+zx)=0
Is the statement true? Give reason for your answer, either with a short proof or a counterexample.