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Chapter 51: Problem 898

Find an equation of the sphere which has the segment joining \(\mathrm{P}_{1}(2,-2,4)\) and \(\mathrm{P}_{2}(4,8,-6)\) for a diameter.

Short Answer

Expert verified
The equation of the sphere with the segment joining P1(2, -2, 4) and P2(4, 8, -6) as its diameter is \((x - 3)^2 + (y - 3)^2 + (z + 1)^2 = 51\).

Step by step solution

01

Calculate the midpoint of the segment

To find the midpoint (which is the center of the sphere) of the segment joining P1(2, -2, 4) and P2(4, 8, -6), we need to compute the average of the x-coordinates, y-coordinates, and z-coordinates of both points. The midpoint, M(x, y, z), can be found using the following formula: M(\( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \)) where P1(\(x_1,y_1,z_1\)) and P2(\(x_2,y_2,z_2\)).
02

Find the center of the sphere

Now, we will find the center, C, of the sphere using the midpoint formula: C(\( \frac{2+ 4}{2}, \frac{-2+8}{2}, \frac{4+(-6)}{2} \)) which gives us: C(\(3, 3, -1\))
03

Calculate the radius of the sphere

Next, we need to find the radius of the sphere. The radius can be calculated by finding the distance between the center, C(3, 3, -1), and either of the points, P1(2, -2, 4) or P2(4, 8, -6). Let's use P1: The distance formula for two points in three-dimensional space is given by: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \) Using C(\(3, 3, -1\)) and P1(\(2,-2,4\)), we get: \(d = \sqrt{(2 - 3)^2 + (-2 - 3)^2 + (4 -(-1))^2} \)
04

Compute the radius

Now, we will compute the radius, r: \(r = \sqrt{(-1)^2 + (-5)^2 + 5^2} = \sqrt{1 + 25 + 25} = \sqrt{51}\)
05

Write the equation of the sphere

Finally, we will write the equation of the sphere using the center C(\(x_c, y_c, z_c\)) and the radius r. The general equation of a sphere is given by: \((x - x_c)^2 + (y - y_c)^2 + (z - z_c)^2 = r^2\) Substituting the values of the center C(\(3,3,-1\)) and the radius \(\sqrt{51}\), we get: \((x - 3)^2 + (y - 3)^2 + (z + 1)^2 = 51\) This is the equation of the sphere with the segment joining P1(2, -2, 4) and P2(4, 8, -6) as its diameter.

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