The values are p = 3 or p = -3 and r = i/3 or r = -i/3, where p and r are the slopes of the perpendicular lines y = px + q and y = rx + s. The condition p * r = -1 is satisfied, confirming the perpendicularity of the lines.
To determine the values of p and q given the equations y = px + q and y = rx + s, where the graphs are perpendicular, we need to use the property that the product of the slopes of two perpendicular lines is equal to -1.
The given equations are y = px + q and y = rx + s. The slopes of these lines are p and r, respectively. According to the perpendicularity condition, we have:
p * r = -1
It's also given that p = 9|r|. Substituting this into the perpendicularity condition, we get:
9|r| * r = -1
Since r cannot be zero (as it is in the denominator), we can simplify this to:
9r^2 = -1
Now, solving for r, we find two possible solutions:
r = i/3 or r = -i/3
Now, using the fact that p = 9|r|, we find the corresponding values for p:
For r = i/3, p = 3.
For r = -i/3, p = -3.
Therefore, the values of p and r are p = 3 (or p = -3) and r = i/3 (or r = -i/3).