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 point M.

Answer

To find the coordinates of point M that divides the line segment CD in the ratio 1:2, we first need to determine the coordinates of points C and D, where the line AB intersects the coordinate axes.

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

Given points:
- A(-5, 2) = (x1, y1)
- B(3, 6) = (x2, y2)

The slope (m) of the line through points A and B is given by:
\[ m = \frac{y2 - y1}{x2 - x1} = \frac{6 - 2}{3 - (-5)} = \frac{4}{8} = \frac{1}{2} \]

Using the point-slope form of the line equation:
\[ y - y1 = m(x - x1) \]
Substituting the values:
\[ y - 2 = \frac{1}{2}(x + 5) \]

### Step 2: Simplify the equation of the line.

Multiplying both sides by 2 to eliminate the fraction:
\[ 2(y - 2) = x + 5 \]
\[ 2y - 4 = x + 5 \]
Rearranging gives:
\[ x - 2y + 9 = 0 \]

### Step 3: Find the x-intercept (C) by setting y = 0.

Substituting y = 0 into the equation:
\[ x - 2(0) + 9 = 0 \implies x + 9 = 0 \implies x = -9 \]
Thus, the coordinates of point C are:
\[ C(-9, 0) \]

### Step 4: Find the y-intercept (D) by setting x = 0.

Substituting x = 0 into the equation:
\[ 0 - 2y + 9 = 0 \implies -2y + 9 = 0 \implies 2y = 9 \implies y = \frac{9}{2} \]
Thus, the coordinates of point D are:
\[ D(0, \frac{9}{2}) \]

### Step 5: Use the section formula to find the coordinates of point M.

Point M divides the line segment CD in the ratio 1:2. Using the section formula:
\[ M\left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right) \]
where:
- \(C(x_1, y_1) = (-9, 0)\)
- \(D(x_2, y_2) = (0, \frac{9}{2})\)
- \(m = 1\), \(n = 2\)

Substituting the values:
\[ M\left(\frac{1 \cdot 0 + 2 \cdot (-9)}{1 + 2}, \frac{1 \cdot \frac{9}{2} + 2 \cdot 0}{1 + 2}\right) \]
Calculating the x-coordinate:
\[ M_x = \frac{0 - 18}{3} = \frac{-18}{3} = -6 \]
Calculating the y-coordinate:
\[ M_y = \frac{\frac{9}{2} + 0}{3} = \frac{\frac{9}{2}}{3} = \frac{9}{6} = \frac{3}{2} \]

Thus, the coordinates of point M are:
\[ M(-6, \frac{3}{2}) \]

### Final Answer:
The coordinates of point M are \((-6, \frac{3}{2})\).

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