Find all points on the line x = 3 that are 8 units from the point (10,-10).
To find all the points on the line x = 3 that are 8 units away from the point (10,-10), we need to consider the coordinates of these points.
Let's break this down into steps:
Step 1: Understand the problem
The problem asks us to find all the points on the line x = 3 that are 8 units away from the point (10,-10). In other words, we need to find the coordinates (x, y) of these points.
Step 2: Recall the equation of a line
The given line equation is x = 3, which means all points on this line have an x-coordinate of 3. So, if we can find the y-coordinate of these points, we can determine their coordinates.
Step 3: Determine the y-coordinate
Since all points on the line x = 3 have an x-coordinate of 3, we need to find the y-coordinate such that it satisfies the condition of being 8 units away from the point (10,-10).
Step 4: Calculate the distance between two points
To check if a point is 8 units away from another point, we can use the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, let's consider the point (x, y) on the line x = 3 and the given point (10,-10). We can rewrite the distance formula as:
8 = √((3 - 10)^2 + (y - (-10))^2)
Simplifying this equation, we have:
64 = (3 - 10)^2 + (y + 10)^2
Step 5: Solve the equation
Let's solve the equation 64 = (3 - 10)^2 + (y + 10)^2 to find the possible values of y.
64 = (-7)^2 + (y + 10)^2
64 = 49 + (y + 10)^2
15 = (y + 10)^2
Taking the square root of both sides, we get:
±√15 = y + 10
To find the y-coordinate, we solve for y by subtracting 10 from both sides:
y = -10 ± √15
Step 6: Determine the points
Now, we have two possible y-coordinates: y = -10 + √15 and y = -10 - √15. Since the line x = 3 doesn't have a specific range, we can substitute these y-values into the equation x = 3 to get the corresponding points.
For y = -10 + √15:
Point 1: (3, -10 + √15)
For y = -10 - √15:
Point 2: (3, -10 - √15)
So, the two points on the line x = 3 that are 8 units away from the point (10,-10) are (3, -10 + √15) and (3, -10 - √15).