A circle has its center at the origin and a radius of 6. A line has the equation y = -x + 6. How many points of intersection are there between the circle and the line?
Correct: C
The equation of the circle centered at the origin with radius 6 is x^2 + y^2 = 6^2, which is x^2 + y^2 = 36.
Substitute the equation of the line y = -x + 6 into the circle's equation:
x^2 + (-x + 6)^2 = 36
x^2 + (x^2 - 12x + 36) = 36
2x^2 - 12x + 36 = 36
2x^2 - 12x = 0
Factor out 2x:
2x(x - 6) = 0
This gives two possible values for x:
2x = 0 => x = 0
x - 6 = 0 => x = 6
For each x-value, find the corresponding y-value using y = -x + 6:
If x = 0, y = -0 + 6 = 6. So, (0, 6) is an intersection point.
If x = 6, y = -6 + 6 = 0. So, (6, 0) is an intersection point.
Since there are two distinct (x, y) solutions, there are 2 points of intersection.