+4 A line and a circle intersect at and as shown below. Find the coordinates of the midpoint of AB.
[asy]
usepackage("amsmath");
usepackage("amssymb");
unitsize(2 cm);
pair A, B;
A = dir(240);
B = dir(330);
draw(Circle((0,0),1));
draw(interp(A,B,-0.5)--interp(A,B,1.5));
dot("$A$", A, S);
dot("$B$", B, SE);
label("$x^2 + y^2 = 6$", dir(45), NE);
label("$y = \dfrac{x - 4}{2}$", interp(A,B,1.5), E);
[/asy]
+174 Find the intersection points (A and B):
We need to solve the system of equations for x and y:
y = (x - 4)/2 (equation of the line)
x^2 + y^2 = 6 (equation of the circle)
Substitute the first equation for y in the second equation: x^2 + ((x - 4)/2)^2 = 6
Solve for x: x^2 + (x^2 - 8x + 16)/4 = 6 4x^2 + x^2 - 8x + 16 = 24 (multiply both sides by 4 to eliminate the fraction) 5x^2 - 8x - 8 = 0 (x - 2)(5x + 4) = 0
Therefore, x = 2 or x = -4/5.
Find the corresponding y values for each x:
For x = 2: y = (2 - 4)/2 = -1
For x = -4/5: y = (-4/5 - 4)/2 = -14/5
Midpoint coordinates:
The midpoint of points A (2, -1) and B (-4/5, -14/5) is: Midpoint = ((2 - 4/5) / 2, (-1 - 14/5) / 2) Midpoint = (6/5, -9/5)
Therefore, the midpoint of AB is (6/5, -9/5).
