Write an equation of the locus of points
a. equidistant from the x-axis and the y-axis and whose coordinates have the same sign
b.equidistant from the x-axis and the y-axis and whose coordinates have the same sign
Can anyone please give me some ideas to do it? THANKS A LOT!
wouldn't it be a line at a 45 degree angle to the axis?
87
To find the equation of the locus of points equidistant from the x-axis and the y-axis, we need to look at the characteristics shared by all such points.
For part a, we want to find points equidistant from both axes and whose coordinates have the same sign. Let's break the solution down into a step-by-step process:
1. Consider a point (x, y) that satisfies the given conditions.
2. Since the point is equidistant from the x-axis and the y-axis, the distances from the point to the x-axis and the y-axis should be equal. Mathematically, this can be represented as:
|y| = |x| (since the absolute values will ensure that the distances are equal)
3. We also know that the coordinates (x, y) have the same sign, which means both x and y are either positive or negative.
4. Now, if we square both sides of the equation from step 2, we get:
y^2 = x^2
5. Combining step 4 with the condition that the coordinates have the same sign, we can say:
y^2 = x^2 (for x and y ≥ 0)
OR
y^2 = x^2 (for x and y ≤ 0)
Hence, the equation of the locus of points equidistant from the x-axis and the y-axis, with coordinates having the same sign, can be represented as:
a. y^2 = x^2 (for x and y ≥ 0)
b. y^2 = x^2 (for x and y ≤ 0)