# Intercepts Made by a Circle on Axes

### Intercepts Made by a Circle on Axes:

#### 1. Definition:

• X-Intercepts: The points where a circle intersects the x-axis.
• Y-Intercepts: The points where a circle intersects the y-axis.

#### 2. X-Intercepts:

• Definition: The x-intercepts are points where the circle touches or crosses the x-axis.
• Coordinate Form: X-intercepts occur at points $\left(a,0\right)$ and $\left(-a,0\right)$ where $a$ is the distance from the circle's center to the x-intercept.
• Equation Relation: For a circle with equation ${x}^{2}+{y}^{2}={r}^{2}$, the x-intercepts are at $\left(±r,0\right)$.

#### 3. Y-Intercepts:

• Definition: The y-intercepts are points where the circle touches or crosses the y-axis.
• Coordinate Form: Y-intercepts occur at points $\left(0,b\right)$ and $\left(0,-b\right)$ where $b$ is the distance from the circle's center to the y-intercept.
• Equation Relation: For a circle with equation ${x}^{2}+{y}^{2}={r}^{2}$, the y-intercepts are at $\left(0,±r\right)$.

#### 4. Calculation from Circle Equation:

• Standard Form: ${x}^{2}+{y}^{2}={r}^{2}$ represents a circle centered at the origin $\left(0,0\right)$ with radius $r$.
• Using Radius: X-intercepts are at $\left(±r,0\right)$ and y-intercepts at $\left(0,±r\right)$.

#### 5. Circle Equation with Center at $\left(h,k\right)$:

• Shifted Circle Equation: $\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$ represents a circle with center $\left(h,k\right)$ and radius $r$.
• Intercepts Calculation: For this equation, the x-intercepts are at $\left(h±r,k\right)$ and y-intercepts at $\left(h,k±r\right)$.

#### 6. Significance and Applications:

• Geometry and Graphing: Understanding intercepts aids in plotting circles on the Cartesian plane.
• Problem Solving: Intercepts provide valuable information about the circle's position and size.

#### 7. Example:

• Circle Equation: ${x}^{2}+{y}^{2}=25$ represents a circle with radius $r=5$.
• Intercepts Calculation:
• X-Intercepts: At $\left(±5,0\right)$.
• Y-Intercepts: At $\left(0,±5\right)$.