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Point P is on the line y = 5x + 3. The coordinates of point Q are (3,-2)  If M is the midpoint of PQ, then M must lie on a certain line.  What is the equation of this line?

 Jun 10, 2020
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We can parameterize the coordinates of P.

 

\(P = (t, 5t + 3)\) for \(t\in\mathbb R\)

 

We use the formula to calculate the midpoint of PQ.

 

\(M = \left(\dfrac{t + 3}{2}, \dfrac{5t + 3 - 2}{2}\right) = \left(\dfrac{t + 3}{2}, \dfrac{5t +1}{2}\right)\)

 

Then, we let M_x be the x-coordinate of M and M_y be the y-coordinate of M. 

We then find the relationship between M_x and M_y.

 

Consider 10M_x.

 

\(10M_x = 5(t + 3) = 5t + 15 = (5t + 1) + 14 = 2M_y + 14\)

\(10M_x - 2M_y - 14 = 0\\ 5M_x - M_y - 7 = 0\)

 

Therefore the point M must lie on the line \(5x - y - 7 = 0\)

 Jun 11, 2020

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