Find the equation of the line joining the following points, using determinants :
`(1,2)` and `(3,6)`

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 need to set up a determinant that includes the coordinates of the two points and a general point (x, y) on the line. The determinant will be set to zero because the area of the triangle formed by these three points should be zero (as they are collinear).

The determinant can be written as:

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

### Step 2: Calculate the determinant
Now, we will calculate the determinant:

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

This simplifies to:

\[ = 1(6 - y) - 2(3 - x) + (3y - 6x) \]

### Step 3: Simplify the expression
Now, we will simplify the expression obtained from the determinant:

\[ = 6 - y - 6 + 2x + 3y - 6x \]

Combining like terms gives:

\[ = (2x - 6x) + (-y + 3y) + (6 - 6) = -4x + 2y \]

### Step 4: Set the equation to zero
Since we set the determinant equal to zero, we have:

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

### Step 5: Rearranging the equation
We can rearrange the equation to express y in terms of x:

\[ 2y = 4x \]

Dividing both sides by 2 gives:

\[ y = 2x \]

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

\[ \boxed{y = 2x} \]