Finding the Equation of a Line Intercepted Between Axes

The problem asks for the equation of a straight line where the segment between the x and y axes is divided by a specific point (2, 3) in a given ratio (3:2). We can use the intercept form of a line and the section formula to solve this.

Understanding the Intercept Form of a Line

A line that intercepts the x-axis at $(a, 0)$ and the y-axis at $(0, b)$ has the equation in intercept form:

\(\frac{x}{a} + \frac{y}{b} = 1\)

Here, 'a' is the x-intercept and 'b' is the y-intercept.

Applying the Section Formula for Ratio Division

The point (2, 3) divides the segment intercepted between the axes in the ratio 3:2. There are two possible ways the ratio can be applied:

1. The point (2, 3) divides the segment from the x-intercept to the y-intercept in the ratio 3:2.
2. The point (2, 3) divides the segment from the y-intercept to the x-intercept in the ratio 3:2.

Let the x-intercept be $A(a, 0)$ and the y-intercept be $B(0, b)$. The given point is $P(2, 3)$.

Scenario 1: P(2, 3) divides AB in the ratio 3:2

In this case, the point P divides the segment from $A(a, 0)$ to $B(0, b)$ in the ratio $m:n = 3:2$. Using the section formula, the coordinates of P are:

\(x = \frac{n x_1 + m x_2}{m+n}\)

\(y = \frac{n y_1 + m y_2}{m+n}\)

Substituting the values $(x, y) = (2, 3)$, \((x_1, y_1) = (a, 0)\), \((x_2, y_2) = (0, b)\), $m=3$, $n=2$:

For the x-coordinate:

\(2 = \frac{2 \cdot a + 3 \cdot 0}{3+2}\)

\(2 = \frac{2a}{5}\)

\(10 = 2a\)

\(a = 5\)

For the y-coordinate:

\(3 = \frac{2 \cdot 0 + 3 \cdot b}{3+2}\)

\(3 = \frac{3b}{5}\)

\(15 = 3b\)

\(b = 5\)

So, the x-intercept is $(5, 0)$ and the y-intercept is $(0, 5)$. The equation of the line is:

\(\frac{x}{5} + \frac{y}{5} = 1\)

Multiplying the entire equation by 5:

\(x + y = 5\)

Scenario 2: P(2, 3) divides BA in the ratio 3:2

In this case, the point P divides the segment from $B(0, b)$ to $A(a, 0)$ in the ratio $m:n = 3:2$. Using the section formula, with \((x_1, y_1) = (0, b)\), \((x_2, y_2) = (a, 0)\), $m=3$, $n=2$:

For the x-coordinate:

\(2 = \frac{2 \cdot 0 + 3 \cdot a}{3+2}\)

\(2 = \frac{3a}{5}\)

\(10 = 3a\)

\(a = \frac{10}{3}\)

For the y-coordinate:

\(3 = \frac{2 \cdot b + 3 \cdot 0}{3+2}\)

\(3 = \frac{2b}{5}\)

\(15 = 2b\)

\(b = \frac{15}{2}\)

So, the x-intercept is \((\frac{10}{3}, 0)\) and the y-intercept is \((\frac{15}{2}, 0)\). The equation of the line is:

\(\frac{x}{10/3} + \frac{y}{15/2} = 1\)

\(\frac{3x}{10} + \frac{2y}{15} = 1\)

To eliminate the denominators, we find the least common multiple (LCM) of 10 and 15, which is 30. Multiply the entire equation by 30:

\(30 \cdot \frac{3x}{10} + 30 \cdot \frac{2y}{15} = 30 \cdot 1\)

\(9x + 4y = 30\)

Summary of Possible Equations

From the two scenarios, we found two possible equations for the line:

• \(x + y = 5\)
• \(9x + 4y = 30\)

These correspond to one of the given options.

Comparing with Options

Let's look at the provided options to find the pair of equations we derived.

Option 4 lists the two equations we found through our calculations.

Revision Table: Key Concepts for Equation of Line

Additional Information: Straight Lines and Intercepts

Understanding how lines interact with the coordinate axes is fundamental in coordinate geometry. The x-intercept is where the line crosses the x-axis (y=0), and the y-intercept is where it crosses the y-axis (x=0). The segment between these two points is often used in problems involving areas of triangles formed by the line and the axes, or problems like this one involving ratio division.

The section formula is a powerful tool for solving problems involving points that divide line segments. It applies whether the point divides the segment internally (as in this problem, the point is on the segment) or externally. For a ratio m:n, it implies that the distance from the first point to the dividing point is proportional to m, and the distance from the dividing point to the second point is proportional to n.

In this problem, the ratio 3:2 implies that the point (2, 3) is closer to one intercept than the other, depending on which intercept is considered the "start" point of the segment for the ratio calculation. This is why we must consider both possibilities, leading to two distinct line equations.