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The line y = (x - 2)/2 intersects the circle x^2 + y^2 = 8 at A and B. Find the midpoint of AB. Express your answer in the form (x,y).

Guest May 1, 2017
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First, let's find where the line y = (x - 2)/2 intersects the circle x2 + y2  =  8 by substituting the first value for y into the second equation:

--->     x2 + [ (x-2)/2 ]2  =  8

Expand the second term:

--->     x2 + ( x2 - 4x + 4 ) / 4  =  8

Multiply all term by 4:

--->     4x2 + x2 - 4x + 4  =  32

Simplify the left side and subtract 32 from both sides:

--->     5x2 - 4x - 28  =  0

Factor (or use the quadratic formula):

--->     (5x - 14)(x + 2)  =  0

Solve for x and put this value into the linear equation to find the corresponding y-value:

--->     5x - 14  =  0     --->     x = 14/5  =  2.8   and   y  =  0.4

    or   x + 2  =  0         --->     x  =  -2   and  y  =  -2

 

The points of intersection are:  (2.8, 0.4)  and  (-2, -2)

 

Use the midpoint formula to find the midpoint of these two points of intersection:

Their midpoint is:     (0.2, -0.8)

geno3141  May 1, 2017

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