Question

How to decide whether lines p_1 and p_2 are perpendicular on the given set of items below? (a) line `p_1: y=3x+5` line `p_2: y=1/3x+5` (b) line `p_1: 3x+5y=12` line `p_2: 5x+3y=18` (c) line `p_1: 4x-2y=6` line `p_2: 2x+4y=6`

Answer 1

Only in case (c), they are perpendicular.

Explanation:

If two lines of the slope-intercept form `y=m_1x+c_1` and `y=m_2x+c_2` are perpendicular to each other if `m_1m_2=-1`.

If one line is of the form `ax+by+c=0`, the line perpendicular to it will be of the form `bx-ay+k=0`, where `k` is another constant. Note that while coefficients of `x` and `y` have changed, sign of one of them too changes.

Note that slope of first line is `-a/b` and that of second is `b/a` and hence product of slopes is `-1`.

(a) Slope of line `p_1` is `3` and that of `p_2` is `1/3`. As product of slopes is `1`, they are not perpendicular.

(b) In the two lines `3x+5y=12` and `5x+3y=18`, although coefficients of `x` and`y` have changed, but signs have remained same. Hence they are not perpendicular.

(c) In the two lines `4x-2y=6` and `2x+4y=6`, coefficients of `x` and`y` have changed and sign of one of them has changed. Hence they are perpendicular.