The equation to the straight line passing through the intersection of `x/a+y/b=1` and `x/b+y/a=1` and (1,2) is ………..

Answer

To find the equation of the straight line passing through the intersection of the lines \( \frac{x}{a} + \frac{y}{b} = 1 \) and \( \frac{x}{b} + \frac{y}{a} = 1 \), and the point (1, 2), we can follow these steps:

### Step 1: Find the point of intersection of the two lines
We have the equations:
1. \( \frac{x}{a} + \frac{y}{b} = 1 \)
2. \( \frac{x}{b} + \frac{y}{a} = 1 \)

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

From the first equation, we can express \( y \) in terms of \( x \):
\[ y = b(1 - \frac{x}{a}) = b - \frac{bx}{a} \]

Substituting this expression for \( y \) into the second equation:
\[ \frac{x}{b} + \frac{b - \frac{bx}{a}}{a} = 1 \]

Multiplying through by \( ab \) to eliminate the denominators:
\[ ax + b(a - \frac{bx}{a}) = ab \]
\[ ax + ab - bx = ab \]
\[ (a - b)x = 0 \]

Thus, \( x = \frac{ab}{a + b} \).

Now substituting \( x \) back into the first equation to find \( y \):
\[ \frac{\frac{ab}{a + b}}{a} + \frac{y}{b} = 1 \]
\[ \frac{b}{a + b} + \frac{y}{b} = 1 \]
\[ \frac{y}{b} = 1 - \frac{b}{a + b} \]
\[ y = b \left( \frac{a}{a + b} \right) = \frac{ab}{a + b} \]

Thus, the point of intersection is \( \left( \frac{ab}{a + b}, \frac{ab}{a + b} \right) \).

### Step 2: Find the slope of the line passing through the intersection point and (1, 2)
Let the point of intersection be \( P\left( \frac{ab}{a + b}, \frac{ab}{a + b} \right) \) and the point (1, 2) be \( Q(1, 2) \).

The slope \( m \) of the line passing through points \( P \) and \( Q \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - \frac{ab}{a + b}}{1 - \frac{ab}{a + b}} \]

### Step 3: Write the equation of the line in point-slope form
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (1, 2) \) and the slope \( m \):
\[ y - 2 = m(x - 1) \]

### Step 4: Simplify the equation
Now we can simplify this equation to find the required line in standard form.

1. Rearranging gives:
\[ y = m(x - 1) + 2 \]

2. Substitute the value of \( m \) from step 2 into this equation and simplify.

### Final Equation
After simplification, we will arrive at the required equation of the line.

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