Find the length of the chord joining the points in which the straight line
`(x)/(a) + (y)/(b) = 1 `
`x^(2) + y^(2) = r^(2)`

Answer

To find the length of the chord joining the points where the straight line \( \frac{x}{a} + \frac{y}{b} = 1 \) intersects the circle \( x^2 + y^2 = r^2 \), we can follow these steps:

### Step 1: Rewrite the line equation
The line equation can be rewritten in standard form. Multiplying through by \( ab \) gives:
\[ bx + ay = ab \]

### Step 2: Find the distance from the center to the line
The center of the circle is at the origin (0, 0). The distance \( d \) from the center to the line \( bx + ay - ab = 0 \) can be calculated using the formula for the distance from a point to a line:
\[ d = \frac{|0 \cdot b + 0 \cdot a - ab|}{\sqrt{b^2 + a^2}} = \frac{ab}{\sqrt{a^2 + b^2}} \]

### Step 3: Use the Pythagorean theorem
Let \( C \) be the center of the circle, \( M \) be the midpoint of the chord, and \( P \) and \( Q \) be the points where the line intersects the circle. In triangle \( CMP \):
- \( CM \) is the distance from the center to the line, which we found to be \( d = \frac{ab}{\sqrt{a^2 + b^2}} \).
- \( CP \) is the radius \( r \) of the circle.
- \( MP \) is the half-length of the chord we want to find.

Using the Pythagorean theorem:
\[ CP^2 = CM^2 + MP^2 \]
Substituting the known values:
\[ r^2 = \left(\frac{ab}{\sqrt{a^2 + b^2}}\right)^2 + MP^2 \]

### Step 4: Solve for \( MP \)
Rearranging the equation gives:
\[ MP^2 = r^2 - \left(\frac{ab}{\sqrt{a^2 + b^2}}\right)^2 \]
Calculating \( MP \):
\[ MP = \sqrt{r^2 - \frac{a^2b^2}{a^2 + b^2}} \]

### Step 5: Find the length of the chord
The length of the chord \( PQ \) is twice the length of \( MP \):
\[ PQ = 2 \cdot MP = 2 \sqrt{r^2 - \frac{a^2b^2}{a^2 + b^2}} \]

### Final Answer
The length of the chord joining the points where the line intersects the circle is:
\[ \text{Length of the chord} = 2 \sqrt{r^2 - \frac{a^2b^2}{a^2 + b^2}} \]

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