1. Sphere C(-2,1,-3), radius=7. T plane at (4,-2,-1)=? (a) 6x-3y+2z=0 (b) r.(6,3,2)=28 (c) 5x-3y-2z=28 (d) r.(6/7,-3/7,2/7)

2. Sphere (x-1)²+(y-5)²+(z-2)²=49 and planr r.(-2,-6,-3)=0 form which circle?(a) (x-1)²+(y-5)²+(z-2)²=19.53 (b) C(0,0,0), radius=4.42 (c) radius=4.42, C(-5/7,-1/7,-4/7) (d) r²=19.53, C(-1,-1,-1)

Please help me, I really don't understand this sphere stuff!

1. To determine the equation of the plane that intersects with the given sphere, we need to find the equation of a plane in the form ax + by + cz = d, where (a, b, c) is the normal vector of the plane and (x, y, z) are the coordinates of a point on the plane.

Given the center of the sphere C(-2, 1, -3) and the radius of 7, we can determine that the equation of the sphere is (x + 2)^2 + (y - 1)^2 + (z + 3)^2 = 49.

To find the normal vector of the plane, we can subtract the coordinates of the center of the sphere from the point on the plane (4, -2, -1). Therefore, the normal vector is (4 - (-2), -2 - 1, -1 - (-3)) = (6, -3, 2).

To find the equation of the plane, we substitute the coordinates of the point (4, -2, -1) and the normal vector (6, -3, 2) into the equation of a plane. This gives us:

6x - 3y + 2z = d

To determine the value of d, we substitute the coordinates of the point (4, -2, -1) into the equation:

6(4) - 3(-2) + 2(-1) = d
24 + 6 - 2 = d
d = 28

Therefore, the equation of the plane that intersects with the given sphere is:

6x - 3y + 2z = 28

So, the correct option is (a) 6x - 3y + 2z = 0.

2. To determine which circle is formed by the intersection of the given sphere and the plane, we need to substitute the given plane equation into the equation of the sphere.

The equation of the sphere is (x - 1)^2 + (y - 5)^2 + (z - 2)^2 = 49.

Substituting the equation of the plane, r.(-2, -6, -3) = 0, into the equation of the sphere, we get:

(-2 - 1)^2 + (-6 - 5)^2 + (-3 - 2)^2 = 49
(-3)^2 + (-11)^2 + (-5)^2 = 49
9 + 121 + 25 = 49
155 ≠ 49

Since the result is not equal, none of the given options is correct.

Therefore, none of the options (a), (b), (c), or (d) are correct in this case.

If you have any further questions, feel free to ask!