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# help!

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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

#1
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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