Find the coordinates of the points of intersection of the straight lines whose equations are
`(x)/(a) + (y)/(b)=1 and (x)/(b) + (y)/(a)=1`

Answer

To find the coordinates of the points of intersection of the straight lines given by the equations:

1. \(\frac{x}{a} + \frac{y}{b} = 1\)
2. \(\frac{x}{b} + \frac{y}{a} = 1\)

we can follow these steps:

### Step 1: Rewrite the equations in a simpler form
Multiply the first equation by \(ab\) to eliminate the denominators:
\[ b x + a y = ab \quad \text{(Equation 1)} \]

Now, multiply the second equation by \(ab\) as well:
\[ a x + b y = ab \quad \text{(Equation 2)} \]

### Step 2: Express \(y\) in terms of \(x\) from Equation 1
From Equation 1, we can express \(y\):
\[ a y = ab - b x \implies y = \frac{ab - b x}{a} \]

### Step 3: Substitute \(y\) into Equation 2
Now substitute the expression for \(y\) into Equation 2:
\[ a x + b \left(\frac{ab - b x}{a}\right) = ab \]

### Step 4: Simplify the equation
Multiply through by \(a\) to eliminate the fraction:
\[ a^2 x + b(ab - b x) = a b a \]
\[ a^2 x + ab^2 - b^2 x = a^2 b \]

### Step 5: Combine like terms
Rearranging gives:
\[ (a^2 - b^2)x = a^2 b - ab^2 \]

### Step 6: Solve for \(x\)
Factoring out \(ab\) from the right side:
\[ (a^2 - b^2)x = ab(a - b) \]
Thus:
\[ x = \frac{ab(a - b)}{a^2 - b^2} \]

### Step 7: Simplify \(x\)
Using the difference of squares:
\[ x = \frac{ab(a - b)}{(a - b)(a + b)} = \frac{ab}{a + b} \quad \text{(for } a \neq b\text{)} \]

### Step 8: Find \(y\) using the value of \(x\)
Substituting \(x\) back into the expression for \(y\):
\[ y = \frac{ab - b \left(\frac{ab}{a + b}\right)}{a} \]
\[ y = \frac{ab(a + b) - ab}{a(a + b)} = \frac{ab(a)}{a(a + b)} = \frac{ab}{a + b} \]

### Final Result
The coordinates of the point of intersection of the two lines are:
\[ \left(\frac{ab}{a + b}, \frac{ab}{a + b}\right) \]