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# HELP-DUE TODAY!!!!!

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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]

Apr 15, 2024
edited by Hermionegranger  Apr 15, 2024

### 1+0 Answers

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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).

Apr 15, 2024