The co-ordinates of two points P and Q are (2, 6) and (-3, 5) respectively. Find :
the co-ordinates of the point where PQ intersects the x-axis. 

Answer

To find the coordinates of the point where the line segment PQ intersects the x-axis, we will follow these steps:

### Step 1: Identify the coordinates of points P and Q
The coordinates of point P are (2, 6) and the coordinates of point Q are (-3, 5).

### Step 2: Use the two-point form of the equation of a line
The equation of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be expressed as:
\[ \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \]

### Step 3: Substitute the coordinates into the equation
Here, \(x_1 = 2\), \(y_1 = 6\), \(x_2 = -3\), and \(y_2 = 5\). Substituting these values into the equation gives:
\[ \frac{y - 6}{x - 2} = \frac{5 - 6}{-3 - 2} \]

### Step 4: Simplify the right side of the equation
Calculating the right side:
\[ \frac{5 - 6}{-3 - 2} = \frac{-1}{-5} = \frac{1}{5} \]
Thus, the equation becomes:
\[ \frac{y - 6}{x - 2} = \frac{1}{5} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 5(y - 6) = 1(x - 2) \]

### Step 6: Expand both sides
Expanding both sides results in:
\[ 5y - 30 = x - 2 \]

### Step 7: Rearrange the equation to the standard form
Rearranging gives:
\[ 5y = x + 28 \]
or
\[ x - 5y + 28 = 0 \]

### Step 8: Find the intersection with the x-axis
To find where the line intersects the x-axis, we set \(y = 0\):
\[ x - 5(0) + 28 = 0 \]
This simplifies to:
\[ x + 28 = 0 \]
Thus:
\[ x = -28 \]

### Step 9: Write the coordinates of the intersection point
The coordinates of the intersection point where line PQ intersects the x-axis are:
\[ (-28, 0) \]

### Summary of the Solution
The coordinates of the point where PQ intersects the x-axis are \((-28, 0)\).

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