8/10/2019 54-EXAM3

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MATH 54 3rd Long Exam

Directions: Do as indicated, and show complete solutions.

I.

Given spheres with equations 2 2 21 : 6 2 8 1S x y z x y z + + + = and2 2 2

1 : 2 2 4 3S x y z x y z + + + = .

(a)

Express the equations of 1S and 2S in center-radius form.

(b)

Find the midpoint of the centers of the two spheres and the distance between them.

II. Given a curve2

2 19

xy+ = on the xy-plane of the Cartesian space.

(a)

Find the equation of the surface generated by revolving the curve about the x-axis.

(b)

Sketch the graph of the surface and identify the surface.

III. Given a point ( )1,1, 1A and the line through the points ( )1,2,1B and ( )2, 1,1C

(a)

Illustrate the situation. Find the vector BA and its length.

(b)

Find the scalar projection of BA on BC and the vector projection of BA on BC.

(c)

Find the directed distance between the point A and the line .

IV.Given the planes 1 : 2 3 0x y z = and 2 : 3 2 0x y z + = .

(a)

Explain why the two planes are not parallel. Find the angle between the two planes.

(b)

Find the equation of the line which is the intersection of the two planes.

(c) Find the symmetric equation of the lines perpendicular to the plane 1 and containing the point ( )2,1, 1 .

V. Given the lines 11 1

:2 2 1

x y z += =

and 2

2 1:

2 2 1

x y z = =

.

(a)

Explain why the two lines are parallel. Are they coincident? Explain why or why not.

(b) Find the equation of the plane containing the line 1 and the point ( )0,2, 1 .

End of Exam