# Contact of Two Circles

### Contact of Two Circles:

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

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

#### 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.
- $d=\sqrt{({h}_{2}-{h}_{1}{)}^{2}+({k}_{2}-{k}_{1}{)}^{2}}$
- 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.