# 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 $(a,0)$ and $(-a,0)$ 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 $(\pm r,0)$.

#### 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 $(0,b)$ and $(0,-b)$ 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 $(0,\pm r)$.

#### 4. **Calculation from Circle Equation:**

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

#### 5. **Circle Equation with Center at $(h,k)$:**

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

#### 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 $(\pm 5,0)$.
**Y-Intercepts:** At $(0,\pm 5)$.