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#

1 Answer
Mar 21, 2018

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.