Question

A straight line through the point \(\mathrm{A}(3,4)\) is such that its intercept between the axes is bisected at A. It's equation is (a) \(3 x-4 y+7=0\) (b) \(4 \mathrm{x}+3 \mathrm{y}=24\) (c) \(3 x+4 y=25\) (d) \(x+y=7\)

Short answer

The correct equation of the line is \(4x + 3y = 24\) (option b), found by using the midpoint formula to determine the intercepts B(6, 0) and C(0, 8), and then the two-point form of a straight line equation to find the equation of the line.

Step-by-step solution

Find the intercepts of the line

As given, point A(3,4) is the midpoint of the intercepts between the axes. Let's call the intercepts on the x-axis and the y-axis as B(x₁, 0) and C(0, y₁), respectively. Using the midpoint formula, we can find x₁ and y₁ as follows: Midpoint = \(\left(\frac{x₁+x₂}{2},\frac{y₁+y₂}{2}\right)\) Since A is the midpoint of B and C, we can write the coordinates of A as: A(3,4) = \(\left(\frac{x₁+0}{2},\frac{0+y₁}{2}\right)\) Let's solve for x₁ and y₁.

Solve for x₁ and y₁

From the previous step, we have: 3 = \(\frac{x₁}{2}\) and 4 = \(\frac{y₁}{2}\), which gives us: x₁ = 6 and y₁ = 8. Thus, the intercepts B and C are B(6, 0) and C(0, 8), respectively.

Find the equation of the line using the intercepts

We can now find the equation of the line passing through the intercepts B(6, 0) and C(0, 8) using the two-point form of a straight line equation: \(\frac{y-y₂}{x-x₂}=\frac{y₁-y₂}{x₁-x₂}\) Plugging in B as (x₁, y₁) and C as (x₂, y₂), we get: \(\frac{y-8}{x}=\frac{0-8}{6-0}\)

Solve for the equation of the line and verify the given options

From the previous step, we have: \(\frac{y-8}{x}=\frac{-8}{6}\) Multiplying both sides by 6x, we get: 6(y - 8) = -8x Expanding and simplifying, we get: 6y - 48 = -8x Rearranging the equation, we get: 8x + 6y = 48 Dividing both sides by 2: 4x + 3y = 24 Comparing this equation with the given options, we can see that it matches option (b). Therefore, the correct equation of the line is 4x + 3y = 24.

Key concepts

Midpoint Formula

Understanding the midpoint formula is essential for solving problems involving line segments in Cartesian coordinates. At its core, the midpoint formula \( \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \) provides a simple way to find the exact center point of a line segment given the coordinates of its endpoints.

For instance, if we have a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) the midpoint, \( M \) of the line segment is determined by averaging the x-coordinates for the x-value of \( M \) and the y-coordinates for its y-value. This is crucial when dealing with problems like locating the center of a rectangle, bisecting a line segment, or even determining a point on a straight line that is equidistant from two other known points.

Equation of a Line

The equation of a line is a fundamental concept in coordinate geometry that provides a relationship between the x and y coordinates of any point on the line. One of the most common forms of the line equation is \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) represents the y-intercept, the point where the line crosses the y-axis.

Different forms of the line equation are used depending on the information available, like point-slope form if a point on the line and the slope are known, or the standard form \( Ax + By = C \) for various analytical purposes. The line equation is a powerful tool in various applications, including predicting values, interpreting slope as a rate of change, and solving geometric problems.

Two-Point Form

When two points through which a line passes are known, the two-point form of the equation of a line is extremely valuable. This form is articulated as \( \frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1} \) where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. The formula essentially derives the slope of the line by considering the vertical change (\( y_2 - y_1 \) ) over the horizontal change ( \( x_2 - x_1 \) ) between the two points, and sets up a proportion with the vertical and horizontal distances from an arbitrary point \( (x, y) \) on the line to one of the endpoints.

This form is particularly useful because it does not require the slope to be known beforehand and provides an equation that is sure to pass through the given points. It is a practical tool for connecting dots, both literally and figuratively, within coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe the plane using an ordered pair of numbers. Key elements such as points, line segments, and shapes are all analytically described and manipulated using equations and a coordinate system, typically the Cartesian coordinate system.

This method allows the representation of geometric figures in a numerical way, enabling the use of mathematical strategies to solve geometric problems. In coordinate geometry, the position of any point or shape is determined relative to a reference, usually in the form of an x-y grid. The concepts of slope, intercepts, distances, and midpoints all interact in this system to provide a rich framework for problem-solving and proofs within geometry.

Students use coordinate geometry to translate shapes, find areas, evaluate distances, and more, making it an indispensable part of mathematics education.