Using determinants, find the equation of the line joining the points (1,2) and (3,6):

Options

A. y=2x
B. x=3y
C. y=x
D. 4x-y =5

Answer

To find the equation of the line joining the points (1, 2) and (3, 6) using determinants, we can follow these steps:

### Step 1: Set up the determinant
We will use the determinant formula for the equation of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\). The general form of the equation of the line is given by the determinant:

\[ \begin{vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{vmatrix} = 0 \]

Here, we have the points \((x_1, y_1) = (1, 2)\) and \((x_2, y_2) = (3, 6)\).

### Step 2: Substitute the points into the determinant
Substituting the values into the determinant, we have:

\[ \begin{vmatrix} x & y & 1 \\ 1 & 2 & 1 \\ 3 & 6 & 1 \end{vmatrix} = 0 \]

### Step 3: Calculate the determinant
Now, we will calculate the determinant. The determinant can be expanded as follows:

\[ = x \begin{vmatrix} 2 & 1 \\ 6 & 1 \end{vmatrix} - y \begin{vmatrix} 1 & 1 \\ 3 & 1 \end{vmatrix} + 1 \begin{vmatrix} 1 & 2 \\ 3 & 6 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. \(\begin{vmatrix} 2 & 1 \\ 6 & 1 \end{vmatrix} = (2 \cdot 1) - (1 \cdot 6) = 2 - 6 = -4\)

2. \(\begin{vmatrix} 1 & 1 \\ 3 & 1 \end{vmatrix} = (1 \cdot 1) - (1 \cdot 3) = 1 - 3 = -2\)

3. \(\begin{vmatrix} 1 & 2 \\ 3 & 6 \end{vmatrix} = (1 \cdot 6) - (2 \cdot 3) = 6 - 6 = 0\)

Substituting these values back into the determinant equation:

\[ x(-4) - y(-2) + 0 = 0 \]

This simplifies to:

\[ -4x + 2y = 0 \]

### Step 4: Rearranging the equation
We can rearrange this equation to find the relationship between \(x\) and \(y\):

\[ 2y = 4x \]

Dividing both sides by 2 gives:

\[ y = 2x \]

### Final Result
The equation of the line joining the points (1, 2) and (3, 6) is:

\[ y = 2x \]