If the point (a, b) divides a line between the axes in the ratio 2 : 3, the equation of the line is

Options

A. `ax+by=5`
B. `bx+ay=5`
C. `(2x)/(a)+(3y)/(b)=5`
D. `2ax+3by=5`

Answer

To find the equation of the line that is divided by the point (a, b) in the ratio 2:3, we can follow these steps:

### Step 1: Understand the coordinates of the intercepts
The line intersects the x-axis at point (a, 0) and the y-axis at point (0, b).

### Step 2: Use the section formula
The point (a, b) divides the line segment joining (0, b) and (a, 0) in the ratio 2:3. According to the section formula, if a point divides a line segment joining two points (x1, y1) and (x2, y2) in the ratio m:n, the coordinates of the dividing point (x, y) can be calculated as:

\[ x = \frac{m \cdot x2 + n \cdot x1}{m + n} \]
\[ y = \frac{m \cdot y2 + n \cdot y1}{m + n} \]

### Step 3: Assign values to the section formula
Here, we have:
- (x1, y1) = (0, b)
- (x2, y2) = (a, 0)
- m = 2
- n = 3

Using the section formula for x-coordinate:

\[ a = \frac{2 \cdot 0 + 3 \cdot a}{2 + 3} = \frac{3a}{5} \]

For y-coordinate:

\[ b = \frac{2 \cdot 0 + 3 \cdot b}{2 + 3} = \frac{3b}{5} \]

### Step 4: Solve for a and b
From the equations derived, we can express a and b in terms of the coordinates of the point (a, b):

1. \( a = \frac{3a}{5} \) implies \( 5a = 3a \) which is not possible unless \( a = 0 \).
2. \( b = \frac{3b}{5} \) implies \( 5b = 3b \) which is not possible unless \( b = 0 \).

This indicates we need to find the intercepts based on the ratio provided.

### Step 5: Set up the equation of the line
The general equation of a line in intercept form is given by:

\[ \frac{x}{a} + \frac{y}{b} = 1 \]

### Step 6: Substitute the values of a and b
Since we have established that the point (a, b) divides the line in the ratio 2:3, we can substitute the values of a and b into the equation of the line:

\[ \frac{x}{\frac{5a}{2}} + \frac{y}{\frac{5b}{3}} = 1 \]

### Step 7: Simplify the equation
Multiplying through by the least common multiple (LCM) of the denominators (which is 6):

\[ 6 \left( \frac{x}{\frac{5a}{2}} + \frac{y}{\frac{5b}{3}} \right) = 6 \]

This simplifies to:

\[ \frac{12x}{5a} + \frac{18y}{5b} = 6 \]

### Step 8: Rearranging the equation
Multiply through by \(5ab\):

\[ 12bx + 18ay = 30ab \]

### Final Equation
Thus, the equation of the line is:

\[ 2x + 3y = 5 \]