Question

can someone please answer for me that quaestions. please make sure 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 our graphs have some accuracy, we will project the surfaces onto the 3 coordinate planes. For each equation, draw four separate graphs for the 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 rough understanding of the shape of the curves/ surfaces. For parts i–iii you may draw these curves in R³ (instead of R³ ).]

(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 the spheres intersect

i. no where,

ii. in a circle,

iii. (exactly) in a point.

(c) Suppose the spheres intersect somehow. The location of the coordinate 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 , call them P₁ and P₂ respectively.

(a) In three separate graphs draw both projections of P₁ and P₂ onto the...

i. xy-plane,

ii. xz-plane,

iii. yz-plane.

(b) Verify that their curve of intersection is

r(t) = <√ 2 cos(t), √ 3 sin(t), sin² (t)> .

[Hint: Show that the curves lives on both surfaces.]

(c) Determine the unit tangent vector for the curve r(t) from part (b) at three t-values:

i. t = 0,

ii. t = π/3,

iii. t = π/2.

(d) Use your data from part (c) to show that the curve r(t) cannot live in a plane.

Answer 1