Question

The line PQ has equation 3x – 2y = 12. P is a point with coordinates (6, 3) and Q is the point

(-2, k).
1 Find the value of k.
2 Find the equation of the line through P that is perpendicular to PQ.
The point R has coordinates (2, -5).
3 Find the exact length PR.

Answer 1

1. -9
2. 2x + 3y = 21
3. 4√5

Explanation:

1. Put the point values in the equation where x and y are, then solve.

... 3(-2) -2k = 12

... -6 -2k = 12 . . . . eliminate parentheses

... -2k = 18 . . . . . . .add 6

... k = -9 . . . . . . . . . divide by -2

2. The perpendicular line will have a slope that is the negative reciprocal of the slope of the given line. The equation for it can be found by swapping the x- and y-coefficients and negating one of them. Then, to make the line go through point (h, k), replace the constant with zero and replace x with (x-h) and y with (y-k).

... 2(x -6) +3(y -3) = 0 . . . . . a line through P perpendicular to PQ

... 2x + 3y = 21

3. The Pythagorean theorem is used for finding the lengths of line segments. The difference in coordinates between P and R is ...

... P - R = (6, 3) - (2, -5) = (6-2, 3-(-5)) = (4, 8)

The length of segment PR can be considered to be the length of the hypotenuse of a right triangle with these leg lengths.

... ║PR║ = √(4² +8²) = √80

... ║PR║ = 4√5