A straight line passes through the points A(-5, 2) and B(3,6). It intersects the co-ordinate axes at points C and D . M is a point on AB which divides CD in the ratio 1: 2. Find :
the co-ordinates of points C and D.

Answer

To find the coordinates of points C and D where the line passing through points A(-5, 2) and B(3, 6) intersects the coordinate axes, we will follow these steps:

### Step 1: Find the equation of the line passing through points A and B.

The formula for the equation of a line in two-point form is given by:

\[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \]

Here, we have:
- \(A(x_1, y_1) = (-5, 2)\)
- \(B(x_2, y_2) = (3, 6)\)

Substituting the values into the formula:

\[ \frac{y - 2}{6 - 2} = \frac{x + 5}{3 - (-5)} \]

This simplifies to:

\[ \frac{y - 2}{4} = \frac{x + 5}{8} \]

### Step 2: Cross-multiply to eliminate the fractions.

Cross-multiplying gives us:

\[ 8(y - 2) = 4(x + 5) \]

Expanding both sides:

\[ 8y - 16 = 4x + 20 \]

### Step 3: Rearranging to standard form.

Rearranging the equation to standard form:

\[ 4x - 8y + 36 = 0 \]

This can be simplified to:

\[ x - 2y + 9 = 0 \]

### Step 4: Find the coordinates of point C (where the line intersects the x-axis).

To find point C, we set \(y = 0\):

\[ x - 2(0) + 9 = 0 \]

This simplifies to:

\[ x + 9 = 0 \implies x = -9 \]

Thus, the coordinates of point C are:

\[ C(-9, 0) \]

### Step 5: Find the coordinates of point D (where the line intersects the y-axis).

To find point D, we set \(x = 0\):

\[ 0 - 2y + 9 = 0 \]

This simplifies to:

\[ -2y + 9 = 0 \implies 2y = 9 \implies y = \frac{9}{2} \]

Thus, the coordinates of point D are:

\[ D(0, \frac{9}{2}) \]

### Final Answer:

The coordinates of points C and D are:
- C: (-9, 0)
- D: (0, 4.5)