If lines 𝑦 equals 𝑎𝑥 plus 𝑏 and 𝑦 equals 𝑐𝑥 plus 𝑑 are perpendicular, which of the following products equals negative one? Is it (a) 𝑎 and 𝑐, (b) 𝑎 and 𝑑, (c) 𝑏 and 𝑐, or (d) 𝑏 and 𝑑?
Let’s recall first what it means for two lines to be perpendicular. If line one is perpendicular to line two, then the two lines will intersect at right angles. Suppose these lines have equations given in the slope–intercept form 𝑦 equals 𝑚 one 𝑥 plus 𝑏 one and 𝑦 equals 𝑚 two 𝑥 plus 𝑏 two. Then, we know that if two lines are perpendicular, the product of their slopes, that’s 𝑚 one multiplied by 𝑚 two, is negative one.
Let’s consider then the slopes of these two lines. Comparing each of their equations to the slope–intercept form, 𝑦 equals 𝑚𝑥 plus 𝑏, we see that the slope of the first line 𝑦 equals 𝑎𝑥 plus 𝑏 is 𝑎. And the slope of the second line 𝑦 equals 𝑐𝑥 plus 𝑑 is 𝑐. If the two lines are perpendicular, then 𝑎 multiplied by 𝑐 must be equal to negative one. The question asked which of the following products equals negative one. So our answer is option (a). It’s 𝑎 and 𝑐.
One thing to note is that this property of perpendicular lines that we used wouldn’t apply if the two lines happened to be one horizontal and one vertical. The two lines would still be perpendicular, but the product of their slopes wouldn’t be negative one, because the slope of one line would be zero and the slope of the other would be infinite. In this case, though, the lines would have equations of the form 𝑥 equals some constant 𝑘 one and 𝑦 equals some other constant 𝑘 two.
Whilst it would be possible to write the equation of the line 𝑦 equals 𝑘 two in the form 𝑦 equals 𝑐𝑥 plus 𝑑, with 𝑐 equal to zero, and 𝑑 equal to 𝑘 two, it would not be possible to write the line with equation 𝑥 equals 𝑘 one in the form 𝑦 equals 𝑎𝑥 plus 𝑏. And so for that reason, we know that the two lines we’re working with in this question are not horizontal and vertical lines because their equations were specified as 𝑦 equals 𝑎𝑥 plus 𝑏 and 𝑦 equals 𝑐𝑥 plus 𝑑.
Our answer to the question “which of the following products equals negative one?” is option (a). It’s 𝑎 and 𝑐.