Question

Obtain the equation of the sphere having center on the line x/3= y/2= z/-5 and passing through the points
�(0,-2,-4) and (2,-1,-1)

Answer 1

Answer on Question #75371– Math – Analytic Geometry

Question

Obtain the equation of the sphere having center on the line $x/3 = y/2 = z/-5$ and passing through the points $(0,-2,-4)$ and $(2,-1,-1)$.

Solution

Using the formula of the equation of the sphere

$$(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = R^2,$$

where $(x_0, y_0, z_0)$ — the center of the sphere, $R$ — radius of the sphere.

As the sphere passing through the points $(0,-2,-4)$ and $(2,-1,-1)$ then the distance from the center to each of these points — radius of the sphere. Using the formula of the distance between two points we will find the radius:

$$R = \sqrt{(x_0 - 0)^2 + (y_0 + 2)^2 + (z_0 + 4)^2}$$

or

$$R = \sqrt{(x_0 - 2)^2 + (y_0 + 1)^2 + (z_0 + 1)^2}$$

let's equate the right parts:

$$\sqrt{(x_0 - 0)^2 + (y_0 + 2)^2 + (z_0 + 4)^2} = \sqrt{(x_0 - 2)^2 + (y_0 + 1)^2 + (z_0 + 1)^2}$$

Then

$$\begin{aligned} &(x_0 - 0)^2 + (y_0 + 2)^2 + (z_0 + 4)^2 = (x_0 - 2)^2 + (y_0 + 1)^2 + (z_0 + 1)^2 \\ &x_0^2 + y_0^2 + 4y_0 + 4 + z_0^2 + 8z_0 + 16 = x_0^2 - 4x_0 + 4 + y_0^2 + 2y_0 + 1 + z_0^2 + 2z_0 + 1 = \\ &4x_0 + 2y_0 + 6z_0 + 14 = 0. \end{aligned}$$

As the sphere having center on the line $x/3 = y/2 = z/-5$ then the point $(x_0, y_0, z_0)$ belongs to the line. Thus:

$$\frac{x_0}{3} = \frac{y_0}{2}, \frac{x_0}{3} = -\frac{z_0}{5}.$$

or

$$y_0 = \frac{2x_0}{3}, z_0 = -\frac{5x_0}{3}.$$

Let's substitute in (2):

$$\begin{aligned} &4x_0 + 2 \cdot \frac{2x_0}{3} + 6\left(-\frac{5x_0}{3}\right) + 14 = 0, \\ &\frac{12x_0 + 4x_0 - 30x_0}{3} = -14, \\ &\frac{-14x_0}{3} = -14, x_0 = 3. \end{aligned}$$

Substitute in (3):

$$y_0 = \frac{2 \cdot 3}{3} = 2, z_0 = -\frac{5 \cdot 3}{3} = -5,$$

then the point $(3,2,-5)$ the center of the sphere

Substitute in (1):

$$R = \sqrt {(3 - 0) ^ {2} + (2 + 2) ^ {2} + (- 5 + 4) ^ {2}} = \sqrt {2 6}$$

Thus the equation of the sphere is

$$(x - 3) ^ {2} + (y - 2) ^ {2} + (z + 5) ^ {2} = 2 6.$$

Answer: $(x - 3)^2 + (y - 2)^2 + (z + 5)^2 = 26$ .

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