Analytic Geometry Answers

Find the vector equation of the plane determined by the points (1, 0, −1), (0, 1, 1) and (−1, 1, 0). Also nd the point of intersection of the line

Show that the closed sphere with centre )7,3,2( and radius 10 in R³ is contained in the

open cube P = {(x, y,z :) x − 2 <11, y − 3 <11, z − 7 <11}.

Does there exist a plane targent to x²-2y²-2z²=8 and which passes through

2x+3y+2z = 8, x−y+2z = 5? Justify your answer.

Find the equation of the plane passing through the line of intersection of the planes

2x+3y+z = 4 and x+y+z = 2, and which is perpendicular to the plane

2x+3y−z = 3.

Does there pass a plane through the lines x+4/3 =

y/2 =

z−1/3

and x/2 =

y−1/1 =

z+1/1

?

Justify.

Find the equation of the right circular cone whose vertex is (1,0,1), the axis is

x−1 = y−2 = z−3, and the semi-vertical angle is 30◦

. Also, find the section of

the cone by the coordinate planes.

Reduce the following equations to standard form, and then identify which conicoids

they represent. Further, give a rough sketch of each.

i) x^2 +y^2 +2x−y−z+3 = 0

ii) 3y^2 +3z^2 +4x+3y+z = 9

Show that x = y = z+1 is a secant line of the sphere x

2 +y

2 +z

2 −x−y+z−1 = 0.

Also find the intercept made by the sphere on the line.

A right circular cylinder passes through the point (1,−1,4) and has the axis along

the line x−1/2 =

y−3/5 =

z+1/3

. Is this information sufficient to determine the equation of

the cylinder? If it is, determine the equation of the cylinder. Otherwise, state

another condition so that the equation can be determined uniquely, and also find the

equation.

Find the equation of the cone with the vertex at (1,−1,2) and the base curve as

(z+1)

2 = x+2, y = 3.