# Common Chord of Two Circles

### Common Chord of Two Circles in Coordinate Geometry:

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

- Two circles with centers $({h}_{1},{k}_{1})$ and $({h}_{2},{k}_{2})$ and radii ${r}_{1}$ and ${r}_{2}$ respectively are represented by their equations:
- Circle 1: $(x-{h}_{1}{)}^{2}+(y-{k}_{1}{)}^{2}={r}_{1}^{2}$
- Circle 2: $(x-{h}_{2}{)}^{2}+(y-{k}_{2}{)}^{2}={r}_{2}^{2}$

#### 2. **Common Chord:**

- The common chord of two circles is a straight line segment that is a chord of both circles.

#### 3. **Finding the Common Chord:**

**Step 1: Circle Equations:** Start by obtaining the equations of both circles.
**Step 2: Simultaneous Equations:** Solve the equations simultaneously to find the points of intersection, which represent the endpoints of the common chord.
**Step 3: Determine the Chord Equation:** Use the coordinates of the intersection points to find the equation of the common chord in the form $y=mx+c$.

#### 4. **Midpoint of Common Chord:**

- The midpoint of the common chord lies on the line joining the centers of the circles and is equidistant from both circles.

#### 5. **Length of Common Chord:**

- The distance between the points of intersection (endpoints of the common chord) can be calculated using the distance formula: $d=\sqrt{({x}_{2}-{x}_{1}{)}^{2}+({y}_{2}-{y}_{1}{)}^{2}}$.

#### 6. **Perpendicular Bisector of the Common Chord:**

- The perpendicular bisector of the common chord passes through the midpoint of the chord and is perpendicular to it.