Let `Delta ABC` is an isosceles trianlge with `AB = AC`. If `B = (0,a) , C = (2a, 0)` and the equation of AB is `3x - 4y + 4a = 0`, then the equation of side AC is

Options

A. `y = 8x - 16 a`
B. `3y = 4x - 8a`
C. `x = 2a`
D. `y + 8x = 16 a`

Answer

To find the equation of side AC in the isosceles triangle ABC, we will follow these steps:

### Step 1: Identify the coordinates of points B and C
Given:
- Point B = (0, a)
- Point C = (2a, 0)

### Step 2: Set up the coordinates for point A
Since triangle ABC is isosceles with AB = AC, we can denote the coordinates of point A as (x, y).

### Step 3: Write the equation of line AB
The equation of line AB is given as:
\[ 3x - 4y + 4a = 0 \]

### Step 4: Use the distance formula to set up an equation
Since AB = AC, we can use the distance formula to express this equality:
\[ AB = \sqrt{(x - 0)^2 + (y - a)^2} \]
\[ AC = \sqrt{(x - 2a)^2 + (y - 0)^2} \]

### Step 5: Square both distances to eliminate the square root
Setting the two distances equal gives us:
\[ (x - 0)^2 + (y - a)^2 = (x - 2a)^2 + (y - 0)^2 \]

### Step 6: Expand both sides
Expanding both sides:
\[ x^2 + (y^2 - 2ay + a^2) = (x^2 - 4ax + 4a^2 + y^2) \]

### Step 7: Simplify the equation
Cancel \(x^2\) and \(y^2\) from both sides:
\[ -2ay + a^2 = -4ax + 4a^2 \]
Rearranging gives:
\[ 4ax - 2ay - 3a^2 = 0 \]

### Step 8: Divide by a (assuming a ≠ 0)
\[ 4x - 2y - 3a = 0 \]

### Step 9: Solve for y in terms of x
Rearranging gives:
\[ 2y = 4x - 3a \implies y = 2x - \frac{3a}{2} \]

### Step 10: Find the slope of line AC
The slope of line AC can be determined using the coordinates of points A and C:
- Point C = (2a, 0)
- Point A = (x, y)

The slope \(m\) is given by:
\[ m = \frac{y - 0}{x - 2a} = \frac{2x - \frac{3a}{2}}{x - 2a} \]

### Step 11: Determine the equation of line AC
Since we know that point C lies on the line AC, we can substitute the coordinates of point C into the line equation derived from the slope.

### Final Step: Write the equation of line AC
The equation of line AC can be expressed in the form:
\[ x = 2a \]

Thus, the equation of side AC is:
\[ \boxed{x = 2a} \]