Question

Find the equation of the sphere passing through pleft parenthesis negative 2 comma 3 comma 4 right parenthesis(−2, 3, 4) and qleft parenthesis 8 comma negative 1 comma 5 right parenthesis(8, −1, 5) with its center at the midpoint of pq.

Answer 1

To find the equation of the sphere passing through two given points with its center at the midpoint, you can use the midpoint formula and distance formula. The equation of the sphere can be written in the form (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2. The equation of the sphere is (x - 3)^2 + (y - 1)^2 + (z - 4.5)^2 = 11.2^2.

Step-by-step explanation:

To find the equation of the sphere passing through the points (-2, 3, 4) and (8, -1, 5) with its center at the midpoint of these points, we first need to find the coordinates of the center. The midpoint formula is used to find the coordinates of the center, which is the average of the coordinates of the two points:

x-coordinate of center = (-2 + 8) / 2 = 3
y-coordinate of center = (3 - 1) / 2 = 1
z-coordinate of center = (4 + 5) / 2 = 4.5

Now that we have the coordinates of the center (3, 1, 4.5) and a point on the sphere (-2, 3, 4), we can use the distance formula to find the radius of the sphere:

radius = sqrt((3 - (-2))^2 + (1 - 3)^2 + (4.5 - 4)^2) = sqrt(125.25) = 11.2 (approximately)

Finally, we can write the equation of the sphere in the form (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2, where (h, k, l) are the coordinates of the center and r is the radius:

(x - 3)^2 + (y - 1)^2 + (z - 4.5)^2 = 11.2^2

Answer 2

In this question , the given points are

`P(-2,3,4), Q(8,-1,5)`

Center is at the midpoint of PQ . And to find the midpoint, we will use the following formula

`( (x_(1)+x_(2))/(2), (y_(1) +y_(2))/(2) , (z_(1)+z_(2))/(2) )`

Substituting the points in the formula, we will get

`((-2+8)/(2), (3-1)/(2), (4+5)/(2) )= (3,1,4.5)`

And PQ gives the diameter, and to find the length of diameter, we will use the following formula

`d = \sqrt{ (x_(2) -x_(1))^2 + (y_(2) -y_(1))^2 + (z_(2) - z_(1))^2}`

Substituting the given values, we will get

`d = √( (8+2)^2 + (-1-3)^2 +  (5-4)^2) = √(100 + 16+1) = √(117) = 3 √(13)`

Radius is half of diameter, that is

`r = (3 √(13))/(2)`

Now we use the equation of sphere, which is

`(x-x_(0))^2 + (y-y_(0))^2 + (z-z_(0))^2 = r^2`

`(x-3)^2 + (y-1)^2 + (z-4.5)^2 = ( (3 √(13))/(2) )^2 \\ (x-3)^2 + (y-1)^2 + (z-4.5)^2 = (117)/(4)`