The base BC of a triangle ABC is bisected at the point (a, b) and equation to the sides AB and AC are respectively `ax+by=1` and `bx+ay=1` Equation of the median through A is:

Options

A. `ax-by=ab`
B. `(2b-1)(ax+by)=ab`
C. `(2ab-1)(ax+by-1)=(a^(2)+b^(2)-1)(bx+ay-1)`
D. `bx-ay=1`

Answer

To find the equation of the median through point A in triangle ABC, we can follow these steps:

### Step 1: Identify the equations of the lines
We are given the equations of the lines AB and AC:
1. \( ax + by = 1 \) (Equation 1)
2. \( bx + ay = 1 \) (Equation 2)

### Step 2: Find the coordinates of point A
To find point A, we need to determine the intersection of the two lines (Equation 1 and Equation 2). We can solve these equations simultaneously.

Multiply Equation 1 by \( b \):
\[ b(ax + by) = b \]
This simplifies to:
\[ abx + b^2y = b \] (Equation 3)

Multiply Equation 2 by \( a \):
\[ a(bx + ay) = a \]
This simplifies to:
\[ abx + a^2y = a \] (Equation 4)

### Step 3: Subtract the two equations
Now, we subtract Equation 4 from Equation 3:
\[ (abx + b^2y) - (abx + a^2y) = b - a \]
This simplifies to:
\[ (b^2 - a^2)y = b - a \]

### Step 4: Solve for y
Rearranging gives:
\[ y = \frac{b - a}{b^2 - a^2} \]
Using the difference of squares, we can factor the denominator:
\[ y = \frac{b - a}{(b - a)(b + a)} = \frac{1}{b + a} \]
Thus, we have:
\[ y = \frac{1}{a + b} \]

### Step 5: Substitute y back to find x
Now, substitute \( y = \frac{1}{a + b} \) back into either Equation 1 or Equation 2 to find x. We will use Equation 1:
\[ ax + b\left(\frac{1}{a + b}\right) = 1 \]
This simplifies to:
\[ ax + \frac{b}{a + b} = 1 \]
Rearranging gives:
\[ ax = 1 - \frac{b}{a + b} \]
\[ ax = \frac{(a + b) - b}{a + b} = \frac{a}{a + b} \]
Thus, we have:
\[ x = \frac{1}{a + b} \]

### Step 6: Coordinates of point A
Now we have the coordinates of point A:
\[ A\left(\frac{1}{a + b}, \frac{1}{a + b}\right) \]

### Step 7: Find the midpoint D of BC
Since BC is bisected at point D, we can denote the coordinates of D as:
\[ D(a, b) \]

### Step 8: Find the slope of AD
The slope \( m \) of line AD can be calculated as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{b - \frac{1}{a + b}}{a - \frac{1}{a + b}} \]

### Step 9: Write the equation of line AD
Using point-slope form, the equation of line AD is:
\[ y - b = m(x - a) \]

### Step 10: Simplify the equation
After substituting and simplifying, we can express the final equation of the median AD.

### Final Equation
The equation of the median through A can be expressed as:
\[ 2ab - 1 = a^2 - b^2 \]