SOLUTION: if the lines x-py+ap^2=0 Ad x-qy+aq^2 are perpendicular Show that pq=-1

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Question 1025651: if the lines x-py+ap^2=0
Ad x-qy+aq^2 are perpendicular
Show that pq=-1
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
The shortcut to proving it is to know what to recognize in the standard form equation for a line. The slopes of your given lines in order are 1%2Fp and 1%2Fq. The lines to be perpendicular will require %281%2Fp%29%281%2Fp%29=-1. Multiply left and right members by pq.
.... there, you have it.


--
So you want a little more information for this.
Basic fact is, if product of the slopes of two lines is negative ONE, then the two lines are perpendicular.

Put each equation into slope-intercept form.
x-py%2Bap%5E2=0
-py=-x-ap%5E2
y=%281%2Fp%29x%2Bap%5E2
-
x-qy%2Baq%5E2=0
-qy=-x-aq%5E2
y=%281%2Fq%29x%2Baq%5E2
-
Note the slopes of the two lines are 1%2Fp and 1%2Fq.
The two lines are GIVEN as perpendicular, and this means according to the basic fact about perpendicular lines in the plane, %281%2Fp%29%281%2Fq%29=-1.

%281%2Fp%29%281%2Fq%29%2Apq=-1%2A%28pq%29
1=-1%2Apq

1%2A%28-1%29=-1%2Apq%2A%28-1%29

-1=pq-------done.