Analytic Geometry Answers

find the equation of cone generated by lines drawn through the origin to meet the circle through the points (3,0,0) ,(0,2,0),(0,0,1)

Identify the following conic and hence reduce it to standard form:

36x² + 20xy + 30y² − 70x + 128y + 80 = 0

Identify the following surfaces as an elliptical paraboloid, hyperbolic paraboloid, a hyperboloid of one sheet, a hyperboloid of two sheets, a cone, a circular cylinder, an elliptical cylinder, or a parabolic cylinder, and identify the axis of symmetry as the x-axis, the y-axis, or the z-axis.


a)−4x^2+9y^2−z^2=0

b)9y^2−5x^2−8z^2+1=0

c)5x^2+8y^2−z=0

d)6y^2+6z^2=1

e)−8x^2−4y^2+z^2=1

a.) Find and identify the traces of the quadric surfacex^2+y^2-z^2= 1 and use the traces to classify the graph of the surface.

b.) If we change the equation in part a.) to x^2-y^2+z^2= 1, how is the graph affected?

c.) What if we change the equation in part a.) to x^2+y^2+ 2y-z^2= 0? How is the graph affected in this case?

Find the equations of the spheres which pass through the circle x²+y²+z²=0, x+2y+3z=0 and touch the plane 4x+3y=15

4. What is the distance between the point (1,2,4) and the plane 3x+ 2y+ 6z= 5?


5. What is the distance between the parallel planes ax+by+cz=d1 and ax+by+cz=d2?

You may wish to try picking a point on one plane that you can specify exactly and

working out the distance from that point to the other plane.

3. What is the cosine of the angle between the plane x+y+z= 5 and any of the

coordinate planes? What about the cosine of the angle between x+y+z= 5 and

x+ 2y+ 3z= 5?