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The problem in the image involves finding the equation of circles given different conditions. Let's break it down and solve each part.

### 1. Determine the equation of the circle in the following cases:

#### a. The circle is centered at \( O(0, 0) \) and is tangent to the line \( x = -4 \).

- **Center**: \( (0, 0) \)
- **Tangent to line**: \( x = -4 \), meaning the radius is the distance from the center to the line.

The distance from \( (0, 0) \) to the line \( x = -4 \) is the absolute value of \( -4 \), which is 4. Hence, the radius \( r = 4 \).

The equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \( (h, k) \) is the center and \( r \) is the radius. Substituting \( h = 0 \), \( k = 0 \), and \( r = 4 \):
\[
x^2 + y^2 = 16
\]

#### b. The circle is centered at \( P(1, -4) \) and passes through the point \( (-3, 2) \).

- **Center**: \( P(1, -4) \)
- **Passes through**: \( (-3, 2) \)

First, calculate the radius by finding the distance between \( P(1, -4) \) and \( (-3, 2) \). The distance formula is:
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting \( (x_1, y_1) = (1, -4) \) and \( (x_2, y_2) = (-3, 2) \):
\[
r = \sqrt{(-3 - 1)^2 + (2 + 4)^2} = \sqrt{(-4)^2 + (6)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}
\]

Now, the equation of the circle with center \( (1, -4) \) and radius \( r = 2\sqrt{13} \) is:
\[
(x - 1)^2 + (y + 4)^2 = (2\sqrt{13})^2
\]
\[
(x - 1)^2 + (y + 4)^2 = 52
\]

### Final Answers:
- **Part a**: The equation is \( x^2 + y^2 = 16 \).
- **Part b**: The equation is \( (x - 1)^2 + (y + 4)^2 = 52 \).

Would you like more details or explanations on any part of this?

### Related Questions:
1. How do you find the equation of a circle when the radius is not directly given?
2. What is the formula for the distance between a point and a line?
3. How would the equation change if the center of the circle moved?
4. What is the geometric interpretation of a circle being tangent to a line?
5. How do you generalize the circle equation for different conic sections?

### Tip:
To check if a point lies on a circle, substitute the coordinates of the point into the circle’s equation and verify if both sides of the equation are equal.

Tentukan persamaan lingkaran berikut. a. Berpusat di titik O(0, 0) dan menyinggung garis x = -4. b. Berpusat di titik P(1, -4) dan melalui titik (-3, 2).

Find the equation of a circle centered at O(0, 0) and tangent to the line x = -4, and the equation of a circle centered at P(1, -4) passing through the point (-3, 2). Learn key geometry concepts and the distance formula in this problem suitable for high school students.

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