Need help! Correct answer will receive thumbs up. Calculus 3 Answer choices: at

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Need help! Correct answer will receive thumbs up. Calculus 3

Answer choices: at zero points, at one point, at two points, in a line, or in a circle.

  

5)2 (y-4) (2-3) (1 point Consider the sphere (z (a) oes the sphere intersect each of the following planes at zero points, at one point, at two points, in a line, or in a circle? The sphere intersects the xz-plane n a circle The sphere intersects the plane n a circle The sphere intersects the yz-plane at one point (b) Does the sphere intersect each of the following coordinate axes at zero points, at one point, at two points, or in a line? The sphere intersects the x-axis at zero points The sphere intersects the z-axis in a line The sphere intersects the y-axis at zero points

Explanation / Answer

a) we have given sphere (x-5)2+(y-4)2+(z-3)2=25

The equation of the xz-plane is y = 0

The coordinates of the points of intersection of the circle and the plane y = 0 are (x,0,z) satisfying

(x-5)2+(0-4)2+(z-3)2=25

(x-5)2+(z-3)2=25-16=9

The sphere intersects the xz-plane in a circle of center (5,0,3) and radius 3 in the plane y = 0

The equation of the xy-plane is z = 0

The coordinates of the points of intersection of the circle and the plane z = 0 are (x,y,0) satisfying

(x-5)2+(y-4)2+(0-3)2=25

(x-5)2+(y-4)2=25-9=16

The sphere intersects the xy-plane in a circle of center (5,4,0) and radius 4 in the plane z = 0

The equation of the yz-plane is x = 0

The coordinates of the points of intersection of the circle and the plane x = 0 are (0,y,z) satisfying

(0-5)2+(y-4)2+(z-3)2=25

(y-4)2+(z-3)2=0

The sphere intersects the yz-plane at the point (0,4,3) (tangent point)

b) The equations of the x-axis are y = 0 and z = 0

The coordinates of the points of intersection, if there exist, of the sphere and the line y = 0, z = 0 are (x,0,0) satisfying

(x-5)2+(y-4)2+(z-3)2=25

(x-5)2+(0-4)2+(0-3)2=25

(x-5)2=0

x=5

The sphere and the x-axis intersect (tangent) at one point (5,0,0)

The equations of the z-axis are x = 0 and y = 0

The coordinates of the points of intersection, if there exist, of the sphere and the line x = 0, y = 0 are (0,0,z) satisfying

(x-5)2+(y-4)2+(z-3)2=25

(0-5)2+(0-4)2+(z-3)2=25

(z-3)2=-16

There are no real z's satisfying that equatin. The sphere and the z-axis do not intersect

The equations of the y-axis are x = 0 and z = 0

The coordinates of the points of intersection, if there exist, of the sphere and the line x = 0, z = 0 are (0,y,0) satisfying

(x-5)2+(y-4)2+(z-3)2=25

(0-5)2+(y-4)2+(0-3)2=25

(y-4)2=-9

There are no real y's satisfying that equatin. The sphere and the y-axis do not intersect

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