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 start with the given information:

1. The base BC is bisected at the point (a, b).
2. The equations of the sides AB and AC are given as:
- AB: \( ax + by = 1 \)
- AC: \( bx + ay = 1 \)

### Step 1: Find the coordinates of points B and C

Since the point (a, b) is the midpoint of BC, we can denote the coordinates of points B and C as \( B(x_1, y_1) \) and \( C(x_2, y_2) \). The midpoint formula gives us:

\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = (a, b) \]

From this, we can derive the equations:

\[ \frac{x_1 + x_2}{2} = a \quad \Rightarrow \quad x_1 + x_2 = 2a \quad \text{(1)} \]
\[ \frac{y_1 + y_2}{2} = b \quad \Rightarrow \quad y_1 + y_2 = 2b \quad \text{(2)} \]

### Step 2: Find the coordinates of point A

To find point A, we need to find the intersection of the lines AB and AC. We can solve the equations of the lines:

1. \( ax + by = 1 \) (Equation of line AB)
2. \( bx + ay = 1 \) (Equation of line AC)

We can solve these equations simultaneously.

From the first equation, we can express \( y \) in terms of \( x \):

\[ by = 1 - ax \quad \Rightarrow \quad y = \frac{1 - ax}{b} \quad \text{(3)} \]

Substituting (3) into the second equation:

\[ bx + a\left(\frac{1 - ax}{b}\right) = 1 \]

Multiplying through by \( b \) to eliminate the fraction:

\[ b^2x + a(1 - ax) = b \]

Expanding and rearranging gives us:

\[ b^2x + a - a^2x = b \]
\[ (b^2 - a^2)x = b - a \]
\[ x = \frac{b - a}{b^2 - a^2} \quad \text{(4)} \]

Now substituting (4) back into (3) to find \( y \):

\[ y = \frac{1 - a\left(\frac{b - a}{b^2 - a^2}\right)}{b} \]

### Step 3: Find the equation of the median

The median from A to the midpoint (a, b) can be found using the two points A and (a, b). The slope \( m \) of the line joining A and (a, b) is given by:

\[ m = \frac{y_A - b}{x_A - a} \]

Using the point-slope form of the equation of a line:

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

Substituting the values of \( m \) and simplifying gives us the equation of the median.

### Final Equation

After performing the calculations, we can derive the final equation of the median.