What does it mean when it says to find the locus of points that are equidistant from the x-axis and y-axis?
Locus is Latin and in geometry implies you are looking for the "place" or "path" (our word location from from locus)
so you want the graph of all the points which are the same distance from both the x and the y axes.
Wouldn't point like that be (1,1), (2,2) , (6.12 , 6.12) ..... ?
or the equation: y = x
I'm sorry Reiny but could you rearrange your explanation? Because I'm not sure I understand
the points equidistant from the axes lie on the angle bisector between the axes.
y=x or y = -x
When you are asked to find the locus of points that are equidistant from the x-axis and y-axis, it means you need to determine all the points in a coordinate plane that are at the same distance from both the x-axis (horizontal line) and the y-axis (vertical line). In other words, you are looking for the set of points that have the same distance to both axes.
To find the locus of points, you can use a geometric approach.
Start by considering an arbitrary point (x, y) in the coordinate plane. The distance from (x, y) to the x-axis is |y| (the absolute value of the y-coordinate), while the distance from (x, y) to the y-axis is |x| (the absolute value of the x-coordinate).
Since we want these distances to be equal, we can set up an equation: |y| = |x|.
Now, let's consider the different cases.
1. When both x and y are positive: In this case, the equation becomes y = x. This represents the line y = x in the coordinate plane.
2. When both x and y are negative: Similarly, the equation becomes -y = -x, which is equivalent to y = x. This represents the same line as before: y = x.
3. When x is positive and y is negative: The equation becomes -y = x. By multiplying both sides by -1, we get y = -x. This represents the line y = -x.
4. When x is negative and y is positive: Similarly, the equation becomes y = -x. This represents the line y = -x.
Therefore, the locus of points equidistant from the x-axis and y-axis is the union of the lines y = x and y = -x. These lines intersect at the origin (0, 0) and extend infinitely in both directions.