If `x^(2)+y^(2)=c^(2)` and `x/a+y/b=1` intersect at A and B, then find AB. Hence deduce the condition that the line touches the circle.

Answer

To solve the problem, we need to find the length of the line segment \( AB \) where the circle defined by the equation \( x^2 + y^2 = c^2 \) intersects with the line defined by \( \frac{x}{a} + \frac{y}{b} = 1 \). We will also deduce the condition under which the line touches the circle.

### Step-by-Step Solution:

1. **Equation of the Circle and Line**:
- The equation of the circle is given by:
\[ x^2 + y^2 = c^2 \]
- The equation of the line can be rewritten as:
\[ y = -\frac{b}{a}x + b \]

2. **Finding the Intersection Points**:
- Substitute \( y \) from the line equation into the circle equation:
\[ x^2 + \left(-\frac{b}{a}x + b\right)^2 = c^2 \]
- Expanding this:
\[ x^2 + \left(\frac{b^2}{a^2}x^2 - 2\frac{b^2}{a}x + b^2\right) = c^2 \]
- Combine like terms:
\[ \left(1 + \frac{b^2}{a^2}\right)x^2 - 2\frac{b}{a}x + (b^2 - c^2) = 0 \]

3. **Using the Quadratic Formula**:
- The quadratic equation in the form \( Ax^2 + Bx + C = 0 \) gives us:
- \( A = 1 + \frac{b^2}{a^2} \)
- \( B = -2\frac{b}{a} \)
- \( C = b^2 - c^2 \)
- The roots (intersection points) can be found using the quadratic formula:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

4. **Finding Length \( AB \)**:
- The distance \( AB \) can be derived from the intersection points. Using the distance formula and the properties of the quadratic roots, we can find:
\[ AB = \frac{2\sqrt{B^2 - 4AC}}{A} \]
- Substituting the values of \( A \), \( B \), and \( C \):
\[ AB = \frac{2\sqrt{\left(-2\frac{b}{a}\right)^2 - 4\left(1 + \frac{b^2}{a^2}\right)(b^2 - c^2)}}{1 + \frac{b^2}{a^2}} \]

5. **Condition for Tangency**:
- For the line to touch the circle, the discriminant of the quadratic equation must be zero:
\[ B^2 - 4AC = 0 \]
- This leads to:
\[ \left(-2\frac{b}{a}\right)^2 - 4\left(1 + \frac{b^2}{a^2}\right)(b^2 - c^2) = 0 \]
- Simplifying this condition will yield the required relationship between \( a \), \( b \), and \( c \).