Question

The line cuts the \(X\)-axis and \(Y\)-axis at the points \(A\) and \(B\) respectively. The point \((5,6)\) divides the line segment \(AB\) internally in the ratio \(3:1\). Then the equation of the line is

• A. \(2x+y=16\)
• B. \(2x+5y=40\)
• C. \(2x-y=4\)
• D. \(2x-5y=-20\)

Hint:

For axis-intercept problems, start with intercept form and then apply the section formula.

Answer 1

The Correct Option is B

Step 1: Assume intercept form of the line.
Let the line cut the axes at \[ A(a,0), \quad B(0,b) \] So, the equation of the line is \[ \frac{x}{a} + \frac{y}{b} = 1 \]
Step 2: Use the section formula.
The point \(P(5,6)\) divides \(AB\) internally in the ratio \(3:1\). Using the section formula: \[ P\left(\frac{3\cdot 0 + 1\cdot a}{3+1},\; \frac{3\cdot b + 1\cdot 0}{3+1}\right) = \left(\frac{a}{4},\frac{3b}{4}\right) \]
Step 3: Equate coordinates.
\[ \frac{a}{4} = 5 \Rightarrow a = 20 \] \[ \frac{3b}{4} = 6 \Rightarrow b = 8 \]
Step 4: Write the equation of the line.
\[ \frac{x}{20} + \frac{y}{8} = 1 \] Multiplying throughout by \(40\): \[ 2x + 5y = 40 \]