Identify the coordinates of Q and R using the midpoint formula and creating two equations for the following case: There are two equations: x - 2y - 4 = 0 and x + y = 5 There is one point: P(1,1) A line is drawn from P to intersect with x - 2y - 4 = 0 at Q and x + y = 5 at R, so that P is the midpoint of QR.

Expert Answers

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Let the line ax+ by + c = 0 join the points Q and R and pass through P.

Let ax + by + c intersect x - 2y-4 = 0 at Q(x1, y1). Let ax + by + c intersect x +y - 5 = 0 at R(x2, y2).

As P is equidistant from these points (x1 + x2)/2 = 1 and (y1 + y2)/2 = 1

So we have x1 - 2y1 - 4 = 0, x2 + y2 - 5 = 0 and x1 + x2 = 2 and y1 + y2 = 2.

We can write x1 - 2y1 - 4 = 0 and 2 - x1 + 2 - y1 - 5 = 0

=> x1 - 2y1 - 4 = 0 and x1 + y1 + 1 =0

subtracting the two

=> 3y1 + 5 = 0

=> y1 = -5/3

x1 = -1 + 5/3 = 2/3

x2 = 2 - 2/3 = 4/3

y2 = 2 + 5/3 = 11/3

Therefore Q is (2/3 , -5/3) and R is (4/3 , 11/3)

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