Analytic Geometry Answers

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.

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

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.

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 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.

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.

What is the new equation of the conic

x^2 +y^2 +4x−2y+3 = 0, when

i) the origin is shifted at (2,−1), followed by a rotation of axes through 45^◦

?

ii) the axes are rotated through 45^◦

, followed by the shifting of the origin at

(2,−1)?

Are the equations in i) and ii) above the same? Why?