# Common Chord of Two Circles

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

#### 1. Equation of Circles:

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

#### 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: $d=\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}}$.

#### 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.