# 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 $\left(x,y\right)$ are coordinates and $r$ is the radius.

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

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

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

• Distance Formula: For a point $\left({x}_{1},{y}_{1}\right)$ and the circle's center $\left(h,k\right)$, the distance is calculated as $\sqrt{\left({x}_{1}-h{\right)}^{2}+\left({y}_{1}-k{\right)}^{2}}$.
• 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 $\left({x}_{1},{y}_{1}\right)$, 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.