Question

The equation of line equidistant from the points \(\mathrm{A}(1,-2)\) and \(\mathrm{B}(3,4)\) and making congruent angles with the coordinate axes is \(\ldots \ldots \ldots\) (a) \(x+y=1\) (b) \(y-x+1=0\) (c) \(\mathrm{y}-\mathrm{x}-1=0\) (d) \(y-x=2\)

Short answer

The equation of the line equidistant from the points A(1,-2) and B(3,4) and making congruent angles with the coordinate axes is \(y + x = 3\). However, this option is not listed among the given choices.

Step-by-step solution

Find the midpoint of A and B

To find the midpoint (M) of points A(1, -2) and B(3, 4), we use the midpoint formula: M(x, y) = \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\) M(x, y) = \(\left(\frac{1+3}{2}, \frac{-2+4}{2}\right)\) M(x, y) = (2, 1)

Use the fact that the slope is -1

As the line makes congruent angles with x and y axes, its slope (m) is -1. An equation for a line can be written in the form: y - y1 = m(x - x1) In this case, we will use the midpoint M(2, 1) as (x1, y1). Plugging in the coordinates and the slope, we get: y - 1 = -1(x - 2)

Simplify the equation of the line

Now we simplify the equation obtained in Step 2: y - 1 = -1(x - 2) y - 1 = -x + 2 y + x = 3

Compare the equation with the given options

Now we compare the equation of our line (y + x = 3) with the given options: (a) x+y=1 -> This does not match the equation. (b) y-x+1=0 -> Rearranging this, we get y + x = 1. This does not match the equation. (c) y-x-1=0 -> Rearranging this, we get y + x = 1. This does not match the equation. (d) y-x=2 -> Rearranging this, we get y + x = 2(x - 1). Setting x = 1 results in y = -1, so this option also doesn't match the equation. None of the given options match the equation of the line we found (y + x = 3), so the correct answer may not be provided within the options.

Key concepts

Midpoint Formula

When we deal with two points on a coordinate plane, sometimes we need to find the point exactly in the middle of them. This point is called the midpoint. The midpoint formula is useful for this purpose. It helps us find a point, which is equidistant from both endpoints of a line segment.
Let's break down the midpoint formula:

• You have two points, for example, A (\(x_1, y_1\)) and B (\(x_2, y_2\)).
• The formula to find the midpoint M is: \[M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]

This formula simply averages the x-coordinates and y-coordinates of the two given points, giving you a new point that lies exactly in the middle. The midpoint is very useful in problems like finding the center of a line, or when constructing the perpendicular bisector. In our exercise, using A(1, -2) and B(3, 4), we found the midpoint M to be (2, 1), which is equidistant from both A and B.

Slope of a Line

The slope of a line is a measure of how steep the line is. It tells us how much the line moves vertically for a given move horizontally. The slope is fundamental in graphing and understanding linear equations.
The concept of slope can be expressed as follows:

• If you have a line through two points, say A (\(x_1, y_1\)) and B (\(x_2, y_2\)), the slope \(m\) is defined by the ratio \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
• A positive slope means the line goes upwards, while a negative slope means the line goes downwards as you move from left to right on the graph.
• A slope of zero means the line is horizontal, and an undefined slope implies a vertical line.

In our case, we were told that the line makes congruent angles with the x and y axes, which gives it a unique slope of -1. This indicates the line descends at a consistent angle, moving one unit down for each unit it moves to the right.

Equation of a Line

An equation of a line can tell us all the points that lie on that line. It is a statement relating x and y coordinates that chart a line on the graph.
There are several forms of linear equations, but the point-slope form is particularly useful. This form is: \[y - y_1 = m(x - x_1)\]

• Here, \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.
• You can plug in any point and the slope to find the equation of the line through that point.

For instance, in our exercise, by using the midpoint (2, 1) and the slope of -1, we arrived at the line’s equation as \(y + x = 3\). This equation succinctly represents all points that the line passes through, given its slope and its position relative to the midpoint.