Find parametric equations for the line segment joining P (2, −1) and Q(3, 1), and usingparameter form of the line segment also find a) the midpoint between P and Q, b) thepoint that is one-fourth of the way from P to Q, c) the point that is three-fourths ofthe way from P to Q

To find the parametric equations for the line segment joining P(2, -1) and Q(3, 1), we can use the parameter form of a line segment.

Let's denote the parameter as t, where t ranges from 0 to 1.

The x-coordinate of the line segment can be expressed as x = (1 - t) * x1 + t * x2, where x1 and x2 are the x-coordinates of P and Q respectively.

Substituting the values, we have x = (1 - t) * 2 + t * 3. Simplifying this equation gives x = 2 + t.

Similarly, the y-coordinate of the line segment can be expressed as y = (1 - t) * y1 + t * y2, where y1 and y2 are the y-coordinates of P and Q respectively.

Substituting the values, we have y = (1 - t) * (-1) + t * 1. Simplifying this equation gives y = -1 + 2t.

Therefore, the parametric equations for the line segment joining P and Q are: x = 2 + t y = -1 + 2t

a) To find the midpoint between P and Q, we can substitute t = 0.5 into the parametric equations. x = 2 + 0.5 = 2.5 y = -1 + 2(0.5) = 0

So, the midpoint between P and Q is M(2.5, 0).

b) To find the point that is one-fourth of the way from P to Q, we can substitute t = 0.25 into the parametric equations. x = 2 + 0.25 = 2.25 y = -1 + 2(0.25) = -0.5

So, the point that is one-fourth of the way from P to Q is N(2.25, -0.5).

c) To find the point that is three-fourths of the way from P to Q, we can substitute t = 0.75 into the parametric equations. x = 2 + 0.75 = 2.75 y = -1 + 2(0.75) = 0.5

So, the point that is three-fourths of the way from P to Q is O(2.75, 0.5).