In what ratio, the line made by joining the points A(-4, -3) and B(5, 2) intersects x-axis ?

Options

A. `3:2`
B. `2:3`
C. `-3:2`
D. `-2:3`

Answer

To find the ratio in which the line joining the points A(-4, -3) and B(5, 2) intersects the x-axis, we can follow these steps:

### Step 1: Find the coordinates of the intersection point with the x-axis
The x-axis is defined by the line where y = 0. We need to find the point on the line AB where y = 0.

### Step 2: Determine the slope of the line AB
The slope (m) of the line joining two points A(x1, y1) and B(x2, y2) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of A and B:
\[ m = \frac{2 - (-3)}{5 - (-4)} = \frac{2 + 3}{5 + 4} = \frac{5}{9} \]

### Step 3: Write the equation of the line in point-slope form
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using point A(-4, -3):
\[ y - (-3) = \frac{5}{9}(x - (-4)) \]
This simplifies to:
\[ y + 3 = \frac{5}{9}(x + 4) \]

### Step 4: Simplify the equation
Distributing the slope:
\[ y + 3 = \frac{5}{9}x + \frac{20}{9} \]
Now, subtract 3 from both sides to isolate y:
\[ y = \frac{5}{9}x + \frac{20}{9} - 3 \]
Convert 3 into a fraction with a denominator of 9:
\[ y = \frac{5}{9}x + \frac{20}{9} - \frac{27}{9} \]
Thus,
\[ y = \frac{5}{9}x - \frac{7}{9} \]

### Step 5: Set y to 0 to find the x-coordinate of the intersection
Set y = 0:
\[ 0 = \frac{5}{9}x - \frac{7}{9} \]
Solving for x:
\[ \frac{5}{9}x = \frac{7}{9} \]
Multiply both sides by 9:
\[ 5x = 7 \]
Thus,
\[ x = \frac{7}{5} \]

### Step 6: Find the ratio in which the x-axis divides the line segment AB
Let the point of intersection be P(7/5, 0). We can use the section formula to find the ratio in which the x-axis divides the line segment AB. If P divides AB in the ratio k:1, then:
\[ \frac{x_P - x_A}{x_B - x_P} = \frac{k}{1} \]
Substituting the values:
\[ \frac{\frac{7}{5} - (-4)}{5 - \frac{7}{5}} = \frac{k}{1} \]

Calculating the left side:
\[ \frac{\frac{7}{5} + \frac{20}{5}}{\frac{25}{5} - \frac{7}{5}} = \frac{\frac{27}{5}}{\frac{18}{5}} = \frac{27}{18} = \frac{3}{2} \]

Thus, the ratio k:1 is 3:2.

### Final Answer
The line joining the points A(-4, -3) and B(5, 2) intersects the x-axis in the ratio 3:2.

---