Question

The equation of the straight line passing through the point \((-5,3)\) such that the portion of the line intercepted between the axes is divided by the point in the ratio \(4:3\) (from the X-axis to the Y-axis) is...

• A. \(9x+20y+105 = 0\)
• B. \(9x-20y-105 = 0\)
• C. \(9x-20y+105 = 0\)
• D. \(9x+20y-105 = 0\)

Hint:

Let the intercepts be \((a,0)\) and \((0,b)\), then use the section formula to place \((-5,3)\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Let the line meet the X-axis at \(A(a,0)\) and the Y-axis at \(B(0,b)\). The point \(P(-5,3)\) divides \(AB\) in the ratio \(4:3\), measured from the X-axis side to the Y-axis side, so \(AP:PB = 4:3\).

Step 2: Use the section formula:
\[ P = \left(\frac{4\cdot0 + 3a}{7},\ \frac{4b + 3\cdot0}{7}\right) = \left(\frac{3a}{7},\ \frac{4b}{7}\right) \]
So \(\frac{3a}{7} = -5\), which gives \(a = -\frac{35}{3}\), and \(\frac{4b}{7} = 3\), which gives \(b = \frac{21}{4}\).

Step 3: Write the line:
Intercept form: \(\frac{x}{a}+\frac{y}{b} = 1\), i.e. \(-\frac{3x}{35} + \frac{4y}{21} = 1\).
Multiply by the LCM \(105\): \(-9x + 20y = 105\), so \(9x - 20y + 105 = 0\).

Step 4: Verify:
Check \(P(-5,3)\): \(9(-5) - 20(3) + 105 = -45 - 60 + 105 = 0\). It lies on the line. Option A has \(+20y\) and cannot pass through \((-5,3)\) with the right intercepts, and options B and D have the wrong constant sign.

Final Answer:
The required line is \(9x-20y+105=0\), option (C). \[ \boxed{9x-20y+105=0} \]