Analytic Geometry Answers

suppose the position vector of X and Y are (1,2,4) and (2,3,5), find the position vector of a point Z that bisect XY in the ratio 2:3

Trace the conicoid represented by x^2+2z^2°y. Also describe its section by planes x=c, for all c belongs to R.

a) Does there exist a plane targent to x

2 −2y

2 +2z

2 = 8 and which passes through

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

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

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

Prove that the equation of the chord joining the points P(ct, c/t) and Q(cT, c/T) on

the rectangular hyperbola xy = c

2

is x + tT y = c(t + T). M is the midpoint of P Q

and P Q meets the x-axis at N. Prove that OM = MN, where O is the origin

Find the rectangular equation for z=r^2 cos (2theta)



Name the surface.

Let A(3,2,1), B(5,0,2), C(3,3,0) and D(1,-6,8) be four points in R3. use vector methods to solve the following.


a) Find the smaller angle between the side AB and AC.

b) Find the shortest distance between D and ABC plane, without finding the equation of the plane

Show that the tangent from the point with vector -2i-3j to the ellipse 4x^2 +9y^2=36 are perpendicular?

a) Find a non-zero unit vector u with terminal point Q(-7,1,4) such that u has the opposite direction as v=(3,2,1).



b) Find a non-zero unit vector u with terminal point Q(-7,1,4) such that u has the same direction as v=(1,2,3).

Find the vector equation of the plane determined by the points (1, 0, −1),

(0, 1, 1) and (−1, 1, 0). Alsofind the point of intersection of the line