Find the equation of the sphere passing through P(-8,7,4) and Q(2,-5,5) with its center at the midpoint of PQ

To find the equation of a sphere passing through two points \( P(-8, 7, 4) \) and \( Q(2, -5, 5) \), with its center at the midpoint of \( PQ \), we can follow these steps:

### 1. Find the midpoint of \( PQ \)
The midpoint \( M \) of a line segment joining two points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]
Substituting the coordinates of \( P \) and \( Q \):
\[
M = \left( \frac{-8 + 2}{2}, \frac{7 + (-5)}{2}, \frac{4 + 5}{2} \right) = \left( \frac{-6}{2}, \frac{2}{2}, \frac{9}{2} \right) = (-3, 1, 4.5)
\]
So, the center \( C \) of the sphere is \( C(-3, 1, 4.5) \).

### 2. Find the radius of the sphere
The radius \( r \) of the sphere is the distance from the center \( C \) to either point \( P \) or \( Q \). We can use the distance formula to find \( r \).

Using point \( P(-8, 7, 4) \):
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
Substituting the coordinates of \( C \) and \( P \):
\[
r = \sqrt{(-8 - (-3))^2 + (7 - 1)^2 + (4 - 4.5)^2}
\]
\[
r = \sqrt{(-5)^2 + 6^2 + (-0.5)^2} = \sqrt{25 + 36 + 0.25} = \sqrt{61.25}
\]
\[
r = \sqrt{\frac{245}{4}} = \frac{\sqrt{245}}{2}
\]

### 3. Write the equation of the sphere
The equation of a sphere with center \( (h, k, l) \) and radius \( r \) is:
\[
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
\]
Substituting the center \( C(-3, 1, 4.5) \) and \( r^2 = 61.25 \):
\[
(x + 3)^2 + (y - 1)^2 + (z - 4.5)^2 = 61.25
\]

So, the equation of the sphere is:
\[
(x + 3)^2 + (y - 1)^2 + (z - 4.5)^2 = 61.25
\]

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How can the midpoint formula be derived geometrically?
2. What is the general form of the equation of a sphere?
3. How can you verify if a point lies on a given sphere?
4. Can the equation of a sphere be expressed in parametric form?
5. How do the concepts of distance and midpoint extend to higher dimensions?

**Tip:** Always check your results by substituting the points back into the equation to ensure they satisfy it!

Learn how to find the equation of a sphere passing through points P(-8,7,4) and Q(2,-5,5) with its center at the midpoint of PQ. This problem covers coordinate geometry concepts, including the midpoint formula and the distance formula. Suitable for high school students studying coordinate geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation