A straight line intersect the x axis at a and the y axis at b. ab is divided internally at c(8,10) in the ratio of 3:4. find the equation of ab.

Options

A. a. x+y=18
B. b. x+y+2=0
C. c. 5x+3y-70=0
D. d. none of these

Answer

To find the equation of the line \( ab \) that intersects the x-axis at point \( A(a, 0) \) and the y-axis at point \( B(0, b) \), and is divided internally at point \( C(8, 10) \) in the ratio \( 3:4 \), we can follow these steps:

### Step 1: Understand the coordinates of points A and B
The point \( A \) is on the x-axis, so its coordinates are \( (a, 0) \).
The point \( B \) is on the y-axis, so its coordinates are \( (0, b) \).

### Step 2: Use the section formula
The coordinates of point \( C \) that divides the line segment \( AB \) in the ratio \( m:n \) can be calculated using the section formula:
\[ C\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Here, \( m = 3 \), \( n = 4 \), \( (x_1, y_1) = (a, 0) \), and \( (x_2, y_2) = (0, b) \).

### Step 3: Set up the equations for the coordinates of C
Using the section formula, we can set up the equations:
1. For the x-coordinate:
\[ 8 = \frac{3 \cdot 0 + 4 \cdot a}{3 + 4} \implies 8 = \frac{4a}{7} \]
2. For the y-coordinate:
\[ 10 = \frac{3 \cdot b + 4 \cdot 0}{3 + 4} \implies 10 = \frac{3b}{7} \]

### Step 4: Solve for a and b
From the first equation:
\[ 8 = \frac{4a}{7} \implies 4a = 56 \implies a = 14 \]

From the second equation:
\[ 10 = \frac{3b}{7} \implies 3b = 70 \implies b = \frac{70}{3} \]

### Step 5: Write the equation of the line
The equation of the line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
Substituting the values of \( a \) and \( b \):
\[ \frac{x}{14} + \frac{y}{\frac{70}{3}} = 1 \]
To simplify, multiply through by \( 70 \):
\[ 5x + 3y = 70 \]

### Step 6: Rearranging the equation
We can also write this equation in standard form:
\[ 5x + 3y - 70 = 0 \]

Thus, the equation of the line \( ab \) is:
\[ 5x + 3y - 70 = 0 \]

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