# Contact of Two Circles

### Contact of Two Circles:

#### 1. Equation of Circles:

• The equation of a circle with center $\left({h}_{1},{k}_{1}\right)$ and radius ${r}_{1}$ is $\left(x-{h}_{1}{\right)}^{2}+\left(y-{k}_{1}{\right)}^{2}={r}_{1}^{2}$.
• Similarly, a circle with center $\left({h}_{2},{k}_{2}\right)$ and radius ${r}_{2}$ is $\left(x-{h}_{2}{\right)}^{2}+\left(y-{k}_{2}{\right)}^{2}={r}_{2}^{2}$.

#### 2. Geometric Relationship:

• No Intersection: If the distance between the centers of the circles is greater than the sum of their radii ($d>{r}_{1}+{r}_{2}$), the circles don't intersect.
• Tangent Circles: When the distance between the centers is equal to the sum of their radii ($d={r}_{1}+{r}_{2}$), the circles touch externally.
• Intersecting Circles: If the distance between the centers is less than the sum of their radii ($d<{r}_{1}+{r}_{2}$), the circles intersect at two distinct points.
• Contained Circle: If one circle lies entirely within the other, the circles are considered to be in contact.

#### 3. Contact Points:

• Solving for Intersection: Equate the distances between the centers and the radii to determine the points of intersection.
• $d=\sqrt{\left({h}_{2}-{h}_{1}{\right)}^{2}+\left({k}_{2}-{k}_{1}{\right)}^{2}}$
• Case 1: $d>{r}_{1}+{r}_{2}$ → No intersection.
• Case 2: $d={r}_{1}+{r}_{2}$ → Tangent circles.
• Case 3: $d<{r}_{1}+{r}_{2}$ → Intersecting circles.

#### 4. Configuration Analysis:

• Exterior Contact: Circles externally touch at a single point.
• Interior Contact: One circle lies completely inside the other, sharing the same center.
• Partial Overlap: Circles partially intersect, sharing a portion of their circumference.