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Chapter 11: Problem 28

Find an equation of the sphere passing through \(P(-4,2,3)\) and \(Q(0,2,7)\) with its center at the midpoint of \(P Q\)

Short Answer

Expert verified
Answer: The equation of the sphere is (x + 2)^2 + (y - 2)^2 + (z - 5)^2 = 8.

Step by step solution

01

Find the midpoint (Center of the sphere)

To find the center of the sphere, we need to determine the midpoint of points \(P\) and \(Q\). To find the midpoint, we use the formula: Midpoint = \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2})\) Now, plugging the coordinates of the given points, \(C = (\frac{-4+0}{2}, \frac{2+2}{2}, \frac{3+7}{2})\) \(C = (-2, 2, 5)\)
02

Find the radius of the sphere

To find the radius, we will calculate the distance between the center, \(C\), and any one of the points, say \(P\). We'll use the distance formula for points \(C (-2, 2, 5)\) and \(P (-4, 2, 3)\): Distance = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\) Plugging in the coordinates, we get: Radius (r) = \(\sqrt{(-4 + 2)^2 + (2 - 2)^2 + (3 - 5)^2} = \sqrt{(2)^2 + (0)^2 + (-2)^2} = \sqrt{8}\)
03

Find the equation of the sphere

Now we have the center, \(C(-2, 2, 5)\), and the radius, \(\sqrt{8}\). We can plug these values into the general equation of a sphere to arrive at our solution: \((x - a)^2 + (y - b)^2 + (z - c)^2 = r^2\) Substitute \(a\), \(b\), \(c\), and \(r\) with our calculated values: \((x - (-2))^2 + (y - 2)^2 + (z - 5)^2 = (\sqrt{8})^2\) \((x + 2)^2 + (y - 2)^2 + (z - 5)^2 = 8\) This is the equation of the sphere that passes through points \(P(-4, 2, 3)\) and \(Q(0, 2, 7)\) with its center, \(C\), at the midpoint of \(PQ\).

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