# Angle of Intersection of Two Circles

### Angle of Intersection of Two Circles:

#### 1. Equation of Circles:

• Circles are defined by equations $\left(x-{h}_{1}{\right)}^{2}+\left(y-{k}_{1}{\right)}^{2}={r}_{1}^{2}$ and $\left(x-{h}_{2}{\right)}^{2}+\left(y-{k}_{2}{\right)}^{2}={r}_{2}^{2}$ with centers $\left({h}_{1},{k}_{1}\right)$ and $\left({h}_{2},{k}_{2}\right)$, 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: $\theta ={\mathrm{tan}}^{-1}\left(m\right)$.

#### 5. Example:

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

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: $\theta ={\mathrm{tan}}^{-1}\left(-\frac{2}{3}\right)\approx -33.6{9}^{\circ }$.