. Which of the following circles would show the locus of points in a plane 1 unit from the intersection of the lines x + y = 2 and x – y = 4?
a. center (4, 2), radius = 1
b. center (3, –1), radius = 1
c. center (3, 1), radius = 1
d. center (0, 0), radius = 1
To find the locus of points that are 1 unit from the intersection of the lines x + y = 2 and x - y = 4, we need to find the intersection point of these lines and then construct a circle with a radius of 1 unit centered at that point.
First, let's solve the system of equations x + y = 2 and x - y = 4 to find the intersection point:
Adding the two equations together, we get:
(x + y) + (x - y) = 2 + 4
2x = 6
x = 3
Now, substitute the value of x into one of the equations to find the value of y:
x + y = 2
3 + y = 2
y = -1
So, the intersection point of the lines is (3, -1).
Now, let's check which of the given circles has a center at (3, -1) and a radius of 1 unit:
a) Center (4, 2), radius = 1: This circle does not have the correct center.
b) Center (3, -1), radius = 1: This circle has the correct center and radius. It is a possible solution.
c) Center (3, 1), radius = 1: This circle does not have the correct center.
d) Center (0, 0), radius = 1: This circle does not have the correct center.
Therefore, the circle that shows the locus of points 1 unit from the intersection of the lines x + y = 2 and x - y = 4 is option b. Center (3, -1), radius = 1.