Question

The line $L_{1}$ joining the two points $(-1, 2)$ and $(3, 6)$ divides the line $L_{2}$ which passes through $(3, -1)$ in the ratio $1 : 3$ internally, then the equation of $L_{2}$ is

• A. $4x - 3y - 9 = 0$
• B. $4x - 3y + 9 = 0$
• C. $4x + 3y - 9 = 0$
• D. $4x + 3y + 9 = 0$

Hint:

In competitive exams, you may sometimes encounter questions with slightly imprecise phrasing. Always look for the most standard mathematical interpretation that makes the problem solvable. Here, recognizing that "a line divides a line" is a typo for "a line divides a line segment" is crucial to proceeding.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The phrasing of the question is slightly ambiguous ("divides the line $L_2$"). A line cannot divide another line in a specific ratio unless it's a line segment. The standard and mathematically sound interpretation of this problem is that the line $L_2$ divides the line segment $L_1$ (joining the points $(-1, 2)$ and $(3, 6)$) in the ratio $1:3$ internally.
Step 2: Key Formula or Approach:
1. Find the point of intersection $P(x, y)$ that divides the line segment joining $A(-1, 2)$ and $B(3, 6)$ in the ratio $m:n = 1:3$ using the section formula: \[ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \] 2. Use the two-point form to find the equation of the line $L_2$ that passes through this point $P(x, y)$ and the given point $Q(3, -1)$. \[ \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \]
Step 3: Detailed Explanation:
Let the points be $A(-1, 2)$ and $B(3, 6)$. Let the intersection point be $P$. This point divides the segment $AB$ in the ratio $1:3$. Using the section formula: \[ x\text{-coordinate of } P = \frac{1 \cdot (3) + 3 \cdot (-1)}{1 + 3} = \frac{3 - 3}{4} = 0 \] \[ y\text{-coordinate of } P = \frac{1 \cdot (6) + 3 \cdot (2)}{1 + 3} = \frac{6 + 6}{4} = \frac{12}{4} = 3 \] So, the point of intersection is $P(0, 3)$. We are given that the line $L_2$ passes through the point $(3, -1)$ and it must also pass through the intersection point $P(0, 3)$ we just found. Now, we find the equation of the line passing through $(0, 3)$ and $(3, -1)$. Using the two-point form: \[ \frac{y - 3}{x - 0} = \frac{-1 - 3}{3 - 0} \] \[ \frac{y - 3}{x} = \frac{-4}{3} \] Cross-multiplying to simplify: \[ 3(y - 3) = -4x \] \[ 3y - 9 = -4x \] Rearranging into standard form ($Ax + By + C = 0$): \[ 4x + 3y - 9 = 0 \]
Step 4: Final Answer:
The equation of the line $L_2$ is $4x + 3y - 9 = 0$, which matches option (C).