Determining the Coordinate of Point K Given Point J and the Distance Between Them

In mathematics, understanding the relationship between points and their coordinates is a fundamental skill. When we are given the coordinate of one point and the distance between that point and another, we can use this information to determine the possible coordinates of the second point.

Let's consider the following scenario:

Suppose point J has a coordinate of -2, and the distance between point J and point K is 4. What are the possible coordinates for point K?

To solve this problem, we can use the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by the formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

In our case, we know that the distance between point J and point K is 4, so we can set the distance formula equal to 4 and solve for the possible coordinates of point K.

Let's start by assigning variables to the known and unknown values:

• Point J has a coordinate of -2, so x1 = -2 and y1 = 0 (since the point is on the x-axis).
• The distance between point J and point K is 4, so d = 4.
• The coordinates of point K are unknown, so we'll represent them as (x2, y2).

Plugging these values into the distance formula, we get:

4 = √((x2 - (-2))^2 + (y2 - 0)^2)
4 = √((x2 + 2)^2 + y2^2)

Now, we can solve this equation to find the possible coordinates of point K.

First, we can square both sides of the equation to get rid of the square root:

16 = (x2 + 2)^2 + y2^2

Expanding the left side, we get:

16 = x2^2 + 4x2 + 4 + y2^2

Rearranging the terms, we get a quadratic equation in terms of x2:

x2^2 + 4x2 - 12 + y2^2 = 0

To solve this equation, we can use the quadratic formula:

x2 = (-4 ± √(4^2 - 4 * 1 * -12)) / (2 * 1)
x2 = (-4 ± √(16 + 48)) / 2
x2 = (-4 ± √64) / 2
x2 = (-4 ± 8) / 2

This gives us two possible solutions for x2:

1. x2 = 2
2. x2 = -6

Now, we can plug these values back into the original distance formula to find the corresponding values of y2:

For x2 = 2:

4 = √((2 - (-2))^2 + y2^2)
4 = √(4 + y2^2)
y2 = ±2

For x2 = -6:

4 = √((-6 - (-2))^2 + y2^2)
4 = √(16 + y2^2)
y2 = ±2

Therefore, the possible coordinates for point K are:

1. (2, 2)
2. (2, -2)
3. (-6, 2)
4. (-6, -2)

In conclusion, when point J has a coordinate of -2 and the distance between J and K is 4, the possible coordinates for point K are (2, 2), (2, -2), (-6, 2), and (-6, -2).