Finding the Equation of a Line Segment with a Given Midpoint Between Axes

The problem asks for the equation of a line segment whose midpoint is given as P(3, 4) and whose endpoints lie on the coordinate axes.

Let the line segment intersect the x-axis at point A and the y-axis at point B. Since point A is on the x-axis, its y-coordinate is 0. Let its coordinates be (a, 0). Since point B is on the y-axis, its x-coordinate is 0. Let its coordinates be (0, b).

The point P(3, 4) is the midpoint of the line segment AB. We can use the midpoint formula to relate the coordinates of A, B, and P.

The midpoint formula states that the coordinates of the midpoint \((x_m, y_m)\) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) are given by:

\((x_m, y_m) = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)

In our case, P(3, 4) is the midpoint, and the endpoints are A(a, 0) and B(0, b). So, we have:

\((3, 4) = \left(\frac{a+0}{2}, \frac{0+b}{2}\right)\)

This gives us two separate equations:

1. The x-coordinate: \(3 = \frac{a+0}{2} \Rightarrow 3 = \frac{a}{2}\)
2. The y-coordinate: \(4 = \frac{0+b}{2} \Rightarrow 4 = \frac{b}{2}\)

Solving these equations for 'a' and 'b':

• From equation 1: \(a = 3 \times 2 = 6\). This means the x-intercept is 6, so point A is (6, 0).
• From equation 2: \(b = 4 \times 2 = 8\). This means the y-intercept is 8, so point B is (0, 8).

Now we have the intercepts of the line on the x-axis and y-axis. We can use the intercept form of the equation of a line, which is:

\(\frac{x}{\text{x-intercept}} + \frac{y}{\text{y-intercept}} = 1\)

Substituting the x-intercept \(a = 6\) and the y-intercept \(b = 8\) into this formula:

\(\frac{x}{6} + \frac{y}{8} = 1\)

To get rid of the denominators and write the equation in the standard form \(Ax + By + C = 0\), we find the least common multiple (LCM) of 6 and 8, which is 24. Multiply the entire equation by 24:

\(24 \left(\frac{x}{6} + \frac{y}{8}\right) = 24(1)\)

\(24 \left(\frac{x}{6}\right) + 24 \left(\frac{y}{8}\right) = 24\)

\(4x + 3y = 24\)

Now, move the constant term to the left side to match the options format:

\(4x + 3y - 24 = 0\)

This is the equation of the line segment with the given midpoint between the axes. Comparing this with the provided options, we find the correct equation.

Revision Table

Additional Information on Line Equations and Midpoints

Understanding different forms of linear equations and the midpoint formula is crucial in coordinate geometry. Here's a bit more detail:

• General Form: \(Ax + By + C = 0\), where A, B, and C are constants and A and B are not both zero. This form is useful for various calculations, including finding intercepts and slopes.
• Slope-Intercept Form: \(y = mx + c\), where 'm' is the slope and 'c' is the y-intercept. This form clearly shows the slope and where the line crosses the y-axis.
• Point-Slope Form: \(y - y_1 = m(x - x_1)\), where 'm' is the slope and \((x_1, y_1)\) is a point on the line. Useful when you know the slope and one point.
• Midpoint Formula Derivation: The midpoint formula is essentially finding the average of the x-coordinates and the average of the y-coordinates of the two endpoints. This makes intuitive sense as the midpoint is exactly halfway between the two points along both the horizontal and vertical directions.
• Line Segment vs. Line: The problem refers to a line segment, but asks for the equation of the "line". In this context, it means the equation of the straight line that contains the segment. The segment simply defines the two points (intercepts) that the line passes through.