# Position of a Point with Respect to a Circle

### Position of a Point with Respect to a Circle:

#### 1. **Circle Equation:**

- The equation of a circle in the Cartesian plane is ${x}^{2}+{y}^{2}={r}^{2}$ where $(x,y)$ are coordinates and $r$ is the radius.

#### 2. **Point-Inside, Outside, or On the Circle:**

**Inside the Circle:** A point $(x,y)$ is inside the circle if ${x}^{2}+{y}^{2}<{r}^{2}$.
**Outside the Circle:** A point $(x,y)$ is outside the circle if ${x}^{2}+{y}^{2}>{r}^{2}$.
**On the Circle:** A point $(x,y)$ lies on the circle if ${x}^{2}+{y}^{2}={r}^{2}$.

#### 3. **Distance from the Circle's Center:**

**Distance Formula:** For a point $({x}_{1},{y}_{1})$ and the circle's center $(h,k)$, the distance is calculated as $\sqrt{({x}_{1}-h{)}^{2}+({y}_{1}-k{)}^{2}}$.
**Comparison to Radius:** If the calculated distance is:
- Less than the radius $r$, the point is inside the circle.
- Greater than $r$, the point is outside the circle.
- Equal to $r$, the point lies on the circle.

#### 4. **Using Inequality to Determine Position:**

**Inequality Test:** For a point $({x}_{1},{y}_{1})$, compare ${x}_{1}^{2}+{y}_{1}^{2}$ with ${r}^{2}$.
**Inside or Outside Check:** If ${x}_{1}^{2}+{y}_{1}^{2}<{r}^{2}$, the point is inside; if ${x}_{1}^{2}+{y}_{1}^{2}>{r}^{2}$, it's outside.
**On the Circle Check:** If ${x}_{1}^{2}+{y}_{1}^{2}={r}^{2}$, the point lies on the circle.

#### 5. **Geometric Interpretation:**

**Visualizing the Circle:** Plotting the circle on the Cartesian plane helps understand the point's position relative to the circle.
**Understanding Quadrants:** Analyzing points in relation to the circle's center and quadrants helps determine their positions.

#### 6. **Applications:**

**Geometry and Trigonometry:** Used in understanding geometric relationships and applying trigonometric concepts.
**Real-world Scenarios:** Applicable in navigation, mapping, and determining object positions.