can someone please answer for me that quaestions. please makesure that i understand your work and handwriting. thank you
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1. We will sketch some quadrics, but in order to make sure ourgraphs have some accuracy, we will project the surfaces onto the 3coordinate planes. For each equation, draw four separate graphs forthe surface S:
i. the projection of S onto the xy-plane,
ii. the projection of S onto the xz-plane,
iii. the projection of S onto the zy-plane,
iv. the graph of S (axes optional).
[Note that I am not looking for works of art—just a roughunderstanding of the shape of the curves/ surfaces. For parts i–iiiyou may draw these curves in R3 (instead ofR3 ).]
(a) Ellipsoid: x2 + 2y2 +3z2 = 6
(b) Paraboloid: z = 2x2 +y2 ? 1
(c) Hyperboloid: x2 + y2? z2 = 1
(d) Hyperbolic Paraboloid: z = 2x2 ?y2 + 1
2. Suppose we have two spheres: x2 + y2 +z2 = 1 and (x ? a)2 + (y ? b)2 +(z ? c)2 = r2 , where r > 0.
(a) Identify the centers and radii for each sphere.
(b) Give an example of values for a, b, c, and r so that thespheres intersect
i. no where,
ii. in a circle,
iii. (exactly) in a point.
(c) Suppose the spheres intersect somehow. The location of thecoordinate axes do not change whether or not the planes intersect,so let’s “move” the spheres to make the equations easier.
x2 + y2 + z2 = 1, x2+ y2 + (z ? c)2= r2 .
Show that their intersection must live in a plane.
3. Suppose we have two paraboloids: z = x2 +y2 ? 2 and z = 4 ? 2x2 ? y2 , callthem P1 and P2 respectively.
(a) In three separate graphs draw both projections ofP1 and P2 onto the...
i. xy-plane,
ii. xz-plane,
iii. yz-plane.
(b) Verify that their curve of intersection is
r(t) = 2 (t)> .
[Hint: Show that the curves lives on both surfaces.]
(c) Determine the unit tangent vector for the curve r(t) frompart (b) at three t-values:
i. t = 0,
ii. t = ?/3,
iii. t = ?/2.
(d) Use your data from part (c) to show that the curve r(t)cannot live in a plane.
