Question

Equation of a straight line which passes through the point P(-1, 2) such that the part of the line between the 2 axes is divided internally by the point P in the ratio 3 : 4 is :

• A. $3x - 8y + 14 = 0$
• B. $3x + 8y - 14 = 0$
• C. $8x - 3y + 14 = 0$
• D. $8x + 3y - 14 = 0$

Hint:

To save time, check which of the given options is satisfied by the coordinates of the point $P(-1, 2)$:
- For Option (C): $8(-1) - 3(2) + 14 = -8 - 6 + 14 = 0$. This instantly identifies the correct line.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We can write the equation of a straight line in intercept form and then use the section formula to locate the division point.
Key Formula or Approach:
1. Intercept form of a straight line:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
2. Section formula for internal division of a line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m : n$:
\[ P(x, y) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right) \]

Step 2: Detailed Explanation:
Let the straight line be $\frac{x}{a} + \frac{y}{b} = 1$.
The intercepts on the axes are:
$A(a, 0)$ on the $x$-axis and $B(0, b)$ on the $y$-axis.
The point $P(-1, 2)$ divides the segment $AB$ internally in the ratio $3 : 4$.
Apply the section formula with $m = 3$ and $n = 4$:
\[ x_P = \frac{3 \cdot (0) + 4 \cdot (a)}{3+4} = \frac{4a}{7} \]
\[ y_P = \frac{3 \cdot (b) + 4 \cdot (0)}{3+4} = \frac{3b}{7} \]
We are given that $P = (-1, 2)$.
Equate the coordinates to solve for $a$ and $b$:
1. Solve for $a$:
\[ \frac{4a}{7} = -1 \implies a = -\frac{7}{4} \]
2. Solve for $b$:
\[ \frac{3b}{7} = 2 \implies b = \frac{14}{3} \]
Substitute the values of $a$ and $b$ back into the intercept equation:
\[ \frac{x}{-7/4} + \frac{y}{14/3} = 1 \implies -\frac{4x}{7} + \frac{3y}{14} = 1 \]
Multiply the entire equation by 14 to eliminate the denominators:
\[ -8x + 3y = 14 \implies 8x - 3y + 14 = 0 \]
Therefore, the equation of the line is $8x - 3y + 14 = 0$.

Step 3: Final Answer
The correct option is (C).