Analytic Geometry Answers

Under what conditions on (alpha) , the spheres x^2 + y^2 + z^2 + (alpha)x - y = 0 and x^2 + Y^2 + z^2 + x +2z +1 = 0 intersect each other at an angle of 45^0.

Let R be the point which divides the line segment joining P(2,1,0) and Q(-1,3,4) in the ratio 1:2 such that PR < PQ.
Find the equation of the line passing through R and parallel to the line x/2=y/1=z/3 .

Find the points of intersection of the conics x^2/4 - y^2/9 = 1 and x^2/6 + y^2/9 = 1.

Find the vertices, eccentricity, foci and asymptotes of the hyperbola x^2/8-y^2/4=1.
Also trace it.
Under what conditions on (lamda) the line x - (lamda)y +2 = 0 will be tangent to this hyperbola?
Explain geometrically.

Obtain the equation of the plane passing through the line (x-1)/2 = -(y+1)/1 = (z-3)/4 and
which is perpendicular to the plane x+2y+z=4

(a) Find the equation of the line which passes through (1,under-root3) and makes an angle 30^0 with the line x - under-root3*y + under-root3 =0.
(b) Find the distance of the line obtained in part (a), from the origin by expressing it in the normal form. Also find the intercepts made by this line on the coordinate axes.

Find the equation of the line which passes through (1,under-root3) and makes an angle 30^0 with the line x - under-root3*y + under-root3 = 0.

Find the equation of the plane which passes through the line of intersection of the planes 2x+y-2z=6 and 2x+3y+6z=5 and makes equal angles with these planes.

Find the angle between the lines x=1,z-y=0 and 2x-y=-1, z=1.

For what value(s) of α, the conicoid x^2+y^2+αz^2+2yz+xy+x+2y+z+3=0 has a unique centre? Give reason for your answer.