The word perpendicular connects the lines’ slopes: when both slopes are defined, their product is . Read each slope from as , turn that relationship into one equation, and use a Desmos regression to find the unknown constant. Check whether a coefficient could make a line vertical before dividing by it.
Hints
- Hint 1
In standard form , the slope is as long as isn’t zero. Use the coefficients of both variables in each equation, not the number on the right.
- Hint 2
If a line’s -coefficient is zero, the line is vertical and has no slope. Check that case separately: is the other line horizontal, as it would need to be for the lines to be perpendicular?
- Hint 3
For two nonvertical perpendicular lines, the slopes multiply to . Write that product first, then clear its denominator to get an equation Desmos can solve for .
Step-by-step
Use the perpendicular slopes
Step 1Find each line’s slope
The slope is the number multiplying when is alone. In , it’s when . Apply that rule to both lines:
- Step 2
Check the vertical case
The second slope would be undefined at . Put that value into both equations: . The first line is not horizontal, so it cannot be perpendicular to the vertical line . So cannot work.
- Step 3
Use the right-angle relationship
Perpendicular slopes multiply to : each is the other’s negative reciprocal. Set the product of these two slopes equal to :
- Step 4
Simplify the slope product
Cancel the two minus signs and reduce :
- Step 5
Clear the denominator
Multiply both sides by , which is safe because was ruled out: . This is the equation for to solve.
- Step 6
Solve for the constant
Type in Desmos. The subscript makes a value Desmos can fit, and tells it to make the sides equal. Under PARAMETERS, Desmos shows . Type on the next line and use the fraction button to see . So the constant is . Choice D.