Two circles, each of radius 5 units, have centers at the origin and at (7, 7),
respectively. What is the y-intercept of the line that contains their common
chord?
To find the y-intercept of the line that contains the common chord of the two circles, we first need to find the equation of that line.
The common chord of two circles passes through the intersection points of those circles. Therefore, we need to find the coordinates of the intersection points.
The first circle has a center at the origin, and a radius of 5 units. Its equation is x^2 + y^2 = 5^2.
The second circle has a center at (7, 7), and a radius of 5 units. Its equation is (x-7)^2 + (y-7)^2 = 5^2.
To find the intersection points, we need to solve the system of equations formed by the two circle equations.
First, let's substitute y^2 in the first equation with (25 - x^2):
x^2 + (25 - x^2) = 5^2
25 = 25
This equation is always true, meaning that all x values satisfy it. Therefore, the first circle intersects the x-axis at all x values.
Now let's solve the second equation:
(x-7)^2 + (y-7)^2 = 5^2
Expanding and simplifying:
x^2 - 14x + 49 + y^2 - 14y + 49 = 25
x^2 + y^2 - 14x - 14y + 73 = 25
x^2 + y^2 - 14x - 14y + 48 = 0
Since this is a quadratic equation in terms of x and y, there are two possible solutions. However, we are only interested in the intersection points that lie on the x-axis. That means y = 0.
Substituting y = 0, the equation becomes:
x^2 - 14x + 48 = 0
Factoring:
(x-4)(x-12) = 0
Solving each factor for x:
x-4 = 0
x = 4
x-12 = 0
x = 12
Therefore, the x-coordinates of the two intersection points are x = 4 and x = 12. Since we are interested in the y-intercept, we need to find the corresponding y-values.
Substituting x = 4 into the equation of the first circle:
4^2 + y^2 = 5^2
16 + y^2 = 25
y^2 = 25 - 16
y^2 = 9
y = ±3
The first intersection point is (4, 3), and the second intersection point is (4, -3).
Substituting x = 12 into the equation of the first circle:
12^2 + y^2 = 5^2
144 + y^2 = 25
y^2 = 25 - 144
y^2 = -119
Since y^2 is negative, there are no real solutions for y. This means that the second intersection point (12, y) does not lie on the x-axis.
Therefore, the common chord of the two circles intersects the x-axis at (4, 0) and has a y-intercept of 0.