Analytic Geometry Answers

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.

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