# Angle of Intersection of Two Circles

### Angle of Intersection of Two Circles:

#### 1. **Equation of Circles:**

- Circles are defined by equations $(x-{h}_{1}{)}^{2}+(y-{k}_{1}{)}^{2}={r}_{1}^{2}$ and $(x-{h}_{2}{)}^{2}+(y-{k}_{2}{)}^{2}={r}_{2}^{2}$ with centers $({h}_{1},{k}_{1})$ and $({h}_{2},{k}_{2})$, and radii ${r}_{1}$ and ${r}_{2}$ respectively.

#### 2. **Geometric Relationship:**

- The angle of intersection between two circles is formed by the tangents drawn from the point of intersection of the circles to their centers.

#### 3. **Properties of Tangents:**

- At the point where the circles intersect, the tangents drawn from this point to each circle are perpendicular to the line joining the centers of the circles.

#### 4. **Calculation of Angle:**

**Step 1: Determine the Slope of the Line Joining Centers:**- Calculate the slope $m$ of the line joining the centers using $m=\frac{{k}_{2}-{k}_{1}}{{h}_{2}-{h}_{1}}$.

**Step 2: Calculate the Angle:**- Use trigonometry to find the angle between the tangents: $\theta ={\mathrm{tan}}^{-1}(m)$.

#### 5. **Example:**

**Circles:**${C}_{1}:(x-2{)}^{2}+(y-3{)}^{2}=4$ and ${C}_{2}:(x+1{)}^{2}+(y-1{)}^{2}=9$.**Centers:**${C}_{1}$ has center $(2,3)$ and ${C}_{2}$ has center $(-1,1)$.

**Calculation:**

**Step 1:**Calculate the slope of the line joining centers: $m=\frac{1-3}{-1-2}=-\frac{2}{3}$.**Step 2:**Determine the angle formed by the tangents: $\theta ={\mathrm{tan}}^{-1}(-\frac{2}{3})\approx -33.6{9}^{\circ}$.