Analytic Geometry Answers

Find the vertices, eccentricity, foci and asymptotes of the hyperbola x²/8-y²/4=1
Also trace it. Under what conditions on λ the line x+λy=2 will be tangent to this hyperbola? Explain geometrically.

Find the equation of the line which passes through the point (1,√3) and makes an angle of 30º with the line x-√(3)y+√3=0

Let R be the point which divides the line segment joining P(2,1,0) and Q(-1,3,4) into 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=z/2

Find the equation of the normal of the paraboloid 3x²+4z²+4y=0 at the point (2,-4,1). Also find the point where this line again intersects the paraboloid.

Find the equation of the cyclinder with base x²+y²+z²+3x+3y-z=1, x+y+2z=2.

Find the equation of the plane which passes through the line of intersection of the planes x+y-2z=1 and 2x+y-4z=3 and which is perpendicular to the plane x+y+z=1.

Find the equation of the tangent plane to the conicoid x²+y²=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.

Find the distance of the point of intersection of the line
x-2/1=y+3/-1=z/3
and the plane 2x-3y+4z+4=0 from the origin.

Identify and trace the conicoid y²+z²=x. Describe its sections by the planes x=0, y=0 and z=0.

Find the transformation of the equation 12x²-2y²+z²=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.