Find the equations of the lines through P(−1, 4) that are (i) parallel and (ii) perpendicular to 3x − 2y + 6 = 0.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found Given: the line 3x − 2y + 6 = 0 and the point P(−1, 4). To find: the lines through P parallel and perpendicular to it.
2 · Draw the figure Sketch the given line and the point P. Draw one line through P parallel to it and another through P at a right angle to it.
3 · Write the rules Parallel to ax + by + c = 0: ax + by + k = 0. Perpendicular: bx − ay + k = 0.
4 · (i) The parallel line
3x − 2y + k = 0
Through P(−1, 4): 3(−1) − 2(4) + k = 0
−3 − 8 + k = 0, so k = 11
3x − 2y + 11 = 0
5 · (ii) The perpendicular line
2x + 3y + k = 0
Through P(−1, 4): 2(−1) + 3(4) + k = 0
−2 + 12 + k = 0, so k = −10
2x + 3y − 10 = 0
6 · Check and box the answer Slopes: 3/2 for the given line and for (i), −2/3 for (ii). (3/2) × (−2/3) = −1. Box both equations.
Scoring tip: After finding k, put P back into your answer. If it does not give 0, check the signs again.
