To find the equations of the circles that pass through the origin and make intercepts of lengths 4 and 8 on the x-axis and y-axis respectively, we can follow these steps:
### Step 1: Understand the Circle's Intercepts
The circle makes intercepts of 4 on the x-axis and 8 on the y-axis. This means:
- The circle intersects the x-axis at points (2, 0) and (-2, 0) (since the total length is 4, half of it is 2).
- The circle intersects the y-axis at points (0, 4) and (0, -4) (since the total length is 8, half of it is 4).
### Step 2: Determine the Center of the Circle
The center of the circle can be determined from the intercepts:
- The center will be at (h, k) where h = ±2 and k = ±4.
Thus, the possible centers of the circle are:
1. (2, 4)
2. (2, -4)
3. (-2, 4)
4. (-2, -4)
### Step 3: Calculate the Radius
The radius \( r \) can be calculated using the distance formula from the origin (0, 0) to the center (h, k):
- For the center (2, 4):
\[
r = \sqrt{(2 - 0)^2 + (4 - 0)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20}
\]
- The same calculation will yield \( r = \sqrt{20} \) for the other centers as well.
### Step 4: Write the Equation of the Circle
The general equation of a circle with center (h, k) and radius r is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Substituting \( r^2 = 20 \) and the centers:
1. For center (2, 4):
\[
(x - 2)^2 + (y - 4)^2 = 20
\]
2. For center (2, -4):
\[
(x - 2)^2 + (y + 4)^2 = 20
\]
3. For center (-2, 4):
\[
(x + 2)^2 + (y - 4)^2 = 20
\]
4. For center (-2, -4):
\[
(x + 2)^2 + (y + 4)^2 = 20
\]
### Step 5: Expand the Equations
1. Expanding the first equation:
\[
(x - 2)^2 + (y - 4)^2 = 20 \implies x^2 - 4x + 4 + y^2 - 8y + 16 = 20 \implies x^2 + y^2 - 4x - 8y = 0
\]
2. Expanding the second equation:
\[
(x - 2)^2 + (y + 4)^2 = 20 \implies x^2 - 4x + 4 + y^2 + 8y + 16 = 20 \implies x^2 + y^2 - 4x + 8y = 0
\]
3. Expanding the third equation:
\[
(x + 2)^2 + (y - 4)^2 = 20 \implies x^2 + 4x + 4 + y^2 - 8y + 16 = 20 \implies x^2 + y^2 + 4x - 8y = 0
\]
4. Expanding the fourth equation:
\[
(x + 2)^2 + (y + 4)^2 = 20 \implies x^2 + 4x + 4 + y^2 + 8y + 16 = 20 \implies x^2 + y^2 + 4x + 8y = 0
\]
### Final Result
The equations of the circles are:
1. \( x^2 + y^2 - 4x - 8y = 0 \)
2. \( x^2 + y^2 - 4x + 8y = 0 \)
3. \( x^2 + y^2 + 4x - 8y = 0 \)
4. \( x^2 + y^2 + 4x + 8y = 0 \)
