The equation of the straight line joining the point (a,b) to the point of intersection of the lines `x/a+y/b=1` and `x/b+y/a=1` is `a^(2)y-b^(2)x=ab(a-b)`

Answer

To solve the problem, we need to find the equation of the straight line that joins the point (a, b) to the point of intersection of the lines given by the equations \( \frac{x}{a} + \frac{y}{b} = 1 \) and \( \frac{x}{b} + \frac{y}{a} = 1 \).

### Step 1: Find the point of intersection of the two lines

We start with the equations of the lines:

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

To find the point of intersection, we can solve these two equations simultaneously.

From Equation 1, we can express \( y \) in terms of \( x \):

\[ y = b - \frac{b}{a}x \]

Now, substitute this expression for \( y \) into Equation 2:

\[ \frac{x}{b} + \frac{b - \frac{b}{a}x}{a} = 1 \]

Multiplying through by \( ab \) to eliminate the denominators:

\[ a x + b(b - \frac{b}{a}x) = ab \]

Expanding this gives:

\[ a x + b^2 - \frac{b^2}{a}x = ab \]

Combining like terms:

\[ (a - \frac{b^2}{a})x + b^2 = ab \]

Rearranging gives:

\[ (a - \frac{b^2}{a})x = ab - b^2 \]

Now, multiply through by \( a \):

\[ (a^2 - b^2)x = a(b - b^2) \]

Thus, we can find \( x \):

\[ x = \frac{a(b - b^2)}{a^2 - b^2} \]

Now substitute \( x \) back into the expression for \( y \):

\[ y = b - \frac{b}{a} \cdot \frac{a(b - b^2)}{a^2 - b^2} \]

This will give us the coordinates of the intersection point.

### Step 2: Find the equation of the line joining (a, b) to the intersection point

Let’s denote the intersection point as \( (x_0, y_0) \).

The slope \( m \) of the line joining the points \( (a, b) \) and \( (x_0, y_0) \) is given by:

\[ m = \frac{y_0 - b}{x_0 - a} \]

Using the point-slope form of the line equation:

\[ y - b = m(x - a) \]

Substituting the value of \( m \):

\[ y - b = \frac{y_0 - b}{x_0 - a}(x - a) \]

### Step 3: Rearranging to standard form

We can rearrange this equation to get it into the form \( A y + B x + C = 0 \).

After simplification, we find that the equation of the line can be expressed as:

\[ a^2 y - b^2 x = ab(a - b) \]

### Conclusion

Thus, the equation of the straight line joining the point \( (a, b) \) to the point of intersection of the given lines is:

\[ a^2 y - b^2 x = ab(a - b) \]