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URL: https://nakafa.com/en/subjects/mathematics/analytic-geometry/position-of-a-point-to-a-circle
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/analytic-geometry/position-of-a-point-to-a-circle/en.mdx

Learn to determine if a point is inside, on, or outside a circle using power of a point method. Learn coordinate geometry with worked examples.

---

## Understanding Point Position Relative to a Circle

If you have a circle and any random point, you must be curious about where that point is, right? Is the point inside the circle, exactly on the edge of the circle, or completely outside the circle?

This concept is really important because in real life we often need to know the position of an object relative to a circular area. For example, is your house still within the signal range of a tower that's circular in shape, or is an airplane's position still within radar surveillance.

By using circle equations and point coordinates, we can determine the position of that point mathematically and accurately.

## Power of a Point Concept

To determine the position of a point relative to a circle, we use a concept called **power of a point**. This is a mathematical way to measure "how far" that point is from the circle.

If we have point $$A(x_0, y_0)$$ and a circle with general equation $$x^2 + y^2 + Dx + Ey + F = 0$$, then the power of point $$A$$ is defined as:

Visible text: If we have point and a circle with general equation , then the power of point is defined as:

```math
K_A = x_0^2 + y_0^2 + Dx_0 + Ey_0 + F
```

So just substitute the point coordinates into the circle equation, easy right?

## Three Possible Point Positions

Based on the power of a point value, there are three possible positions:

1. **Point inside the circle** occurs when $$K_A < 0$$. This means the point is located in the area inside the circle.

2. **Point on the circle** occurs when $$K_A = 0$$. This means the point is exactly located on the edge or circumference of the circle.

3. **Point outside the circle** occurs when $$K_A > 0$$. The point position is located outside the circle area.

Visible text: 1. **Point inside the circle** occurs when . This means the point is located in the area inside the circle.

2. **Point on the circle** occurs when . This means the point is exactly located on the edge or circumference of the circle.

3. **Point outside the circle** occurs when . The point position is located outside the circle area.

If we visualize this, it will look like this:

Component: LineEquation
Props:
- title: Point Position Relative to Circle
- description: Visualization of three possible positions of point $$A$$ relative to the circle.
  Visible text: Visualization of three possible positions of point relative to the circle.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 1 }, () => {
return {
x: 0,
y: 0,
z: 0,
};
}),
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "P(0, 0)", at: 0, offset: [-0.5, -0.5, 0] },
],
},
{
points: Array.from({ length: 1 }, () => {
return {
x: 1.5,
y: 1,
z: 0,
};
}),
color: getColor("TEAL"),
showPoints: true,
labels: [
{ text: "A (inside)", at: 0, offset: [0, -0.5, 0] },
],
},
{
points: Array.from({ length: 1 }, () => {
const angle = Math.PI / 6;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "B (on)", at: 0, offset: [0.5, 0.5, 0] },
],
},
{
points: Array.from({ length: 1 }, () => {
return {
x: 4.5,
y: 2,
z: 0,
};
}),
color: getColor("ROSE"),
showPoints: true,
labels: [
{ text: "C (outside)", at: 0, offset: [0.5, 0.5, 0] },
],
},
{
points: Array.from({ length: 2 }, (_, i) => {
const xMin = -5;
const xMax = 5;
const y = 0;
return {
x: xMin + i * (xMax - xMin),
y: y,
z: 0,
};
}),
color: getColor ... [truncated; 1463 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false

## How to Determine Point Position

The process is quite straightforward. First, we identify the circle equation and coordinates of the point to be checked. Second, substitute the point coordinates into the circle equation to get the power of a point value. Third, look at the sign of the substitution result.

If the result is negative, the point is inside. If zero, the point is on the circle. If positive, the point is outside the circle.

Let's try with a concrete example. Say we have point $$A(1, -2)$$ and circle $$x^2 + y^2 = 25$$.

Visible text: Let's try with a concrete example. Say we have point and circle .

Substitute the point coordinates:

```math
K_A = 1^2 + (-2)^2 = 1 + 4 = 5
```

Since the power of point value $$K_A = 5$$ and for circle $$x^2 + y^2 = 25$$ which has $$r^2 = 25$$, then $$5 < 25$$. So point $$A(1, -2)$$ is located inside the circle.

Visible text: Since the power of point value and for circle which has , then . So point is located inside the circle.

> For circles in the form $$x^2 + y^2 = r^2$$, we compare the substitution result with $$r^2$$. If less than $$r^2$$, point is inside. If equal to $$r^2$$, point is on the circle. If greater than $$r^2$$, point is outside.

Visible text: > For circles in the form , we compare the substitution result with . If less than , point is inside. If equal to , point is on the circle. If greater than , point is outside.

## Application for General Form

If the circle is in general form $$x^2 + y^2 + Dx + Ey + F = 0$$, the method is the same. Just substitute the point coordinates into the entire equation and look at the sign of the result.

Visible text: If the circle is in general form , the method is the same. Just substitute the point coordinates into the entire equation and look at the sign of the result.

For example, for point $$A(1, -2)$$ and circle $$x^2 + y^2 - 8x - 2y - 8 = 0$$:

Visible text: For example, for point and circle :

Component: MathContainer
Children:

```math
K_A = 1^2 + (-2)^2 - 8(1) - 2(-2) - 8
```

```math
K_A = 1 + 4 - 8 + 4 - 8 = -7
```

Since $$K_A = -7 < 0$$, then point $$A(1, -2)$$ is located inside the circle.

Visible text: Since , then point is located inside the circle.

## Practice Problems

1. Determine the position of point $$A(3, 4)$$ relative to circle $$x^2 + y^2 = 16$$.

2. Investigate whether point $$B(6, -8)$$ lies on circle $$x^2 + y^2 = 100$$.

3. Determine the position of point $$C(2, 1)$$ relative to circle $$x^2 + y^2 - 4x + 6y - 12 = 0$$.

4. A circle has equation $$(x-3)^2 + (y+2)^2 = 25$$. Determine the position of point $$D(7, 2)$$ relative to that circle.

Visible text: 1. Determine the position of point relative to circle .

2. Investigate whether point lies on circle .

3. Determine the position of point relative to circle .

4. A circle has equation . Determine the position of point relative to that circle.

### Answer Key

1. **Solution**:

   Substitute coordinates of point $$A(3, 4)$$ into circle equation $$x^2 + y^2 = 16$$:

   
   
   ```math
   K_A = 3^2 + 4^2 = 9 + 16 = 25
   ```

   Since $$K_A = 25 > 16$$, then point $$A(3, 4)$$ is located outside the circle.

2. **Solution**:

   Substitute coordinates of point $$B(6, -8)$$ into circle equation $$x^2 + y^2 = 100$$:

   
   
   ```math
   K_B = 6^2 + (-8)^2 = 36 + 64 = 100
   ```

   Since $$K_B = 100 = 100$$, then point $$B(6, -8)$$ is located exactly on the circle.

3. **Solution**:

   Substitute coordinates of point $$C(2, 1)$$ into circle equation $$x^2 + y^2 - 4x + 6y - 12 = 0$$:

   <MathContainer>
     
   
   ```math
   K_C = 2^2 + 1^2 - 4(2) + 6(1) - 12
   ```

     
   
   ```math
   K_C = 4 + 1 - 8 + 6 - 12 = -9
   ```

   </MathContainer>

   Since $$K_C = -9 < 0$$, then point $$C(2, 1)$$ is located inside the circle.

4. **Solution**:

   For circle $$(x-3)^2 + (y+2)^2 = 25$$ with center $$(3, -2)$$ and radius $$r = 5$$.

   Substitute point $$D(7, 2)$$:

   <MathContainer>
     
   
   ```math
   K_D = (7-3)^2 + (2-(-2))^2 = 4^2 + 4^2
   ```

     
   
   ```math
   K_D = 16 + 16 = 32
   ```

   </MathContainer>

   Since $$K_D = 32 > 25 = r^2$$, then point $$D(7, 2)$$ is located outside the circle.

Visible text: 1. **Solution**:

 Substitute coordinates of point into circle equation :

 
 

 Since , then point is located outside the circle.

2. **Solution**:

 Substitute coordinates of point into circle equation :

 
 

 Since , then point is located exactly on the circle.

3. **Solution**:

 Substitute coordinates of point into circle equation :

 <MathContainer>
 
 

 
 

 </MathContainer>

 Since , then point is located inside the circle.

4. **Solution**:

 For circle with center and radius .

 Substitute point :

 <MathContainer>
 
 

 
 

 </MathContainer>

 Since , then point is located outside the circle.