Analytic Geometry Answers

Trace the conicoid represented by x

2 +2z

2 = y. Also describe its sections by the

planes x = c,∀c ∈ R.

Find the nature of the planar section of the conicoid x

2

3 −

y

2

4 = z by the plane

x+2y−z = 6

Trace the conicoid represented by x²+2z²=y.Also describe its sections by the

planes x= c,∀c ∈ R.

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.

Consider two lines L1 and L2 whose direction cosines l1,m1,n1 and l2,m2,n2 are

given by the equations for l,m,n :

al +bm+cn = 0, f mn+gnl +hlm = 0,

where abc 6= 0. Show that if L1 ⊥ L2, then f/a + g/b +h/c = 0.

Derive the equation (23) at page 42 of Unit 2, which represents the polar equation

of a conic when the directrix L corresponding to a focus F is taken to the right of F.

Suppose a planet describes an ellipse with sun S as its focus, whose major axis is 2a and minor axis is 2b . Let P(x, y) be the position of the planet after time t after starting from rest from perihelion position A , referred to S as origin. Let θ be the eccentric angle of the position P .

Show that

h t = ab (θ − esin θ)

Let A(3,2,-1), B(5,0,-2), C(3,3,0) and D(4,-6,8) be four points in the plane,Donate the plane containing A, B, and C.Use vector methods to solve the following.

(a)Find a unit vector perpendicular to the plane.

(b)WITHOUT finding the equation of the plane , calculate the shortest distance between D and the plane

Find the equation of the line through the origin with slope -6 Answer: 6x+y=0

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