Question

A city planner is rerouting traffic in order to work on a stretch of road. The equation of the path of the old route can be described as y= 2/5 x-4. What should the equation of the new route be if it is to be perpendicular to the old route and will go through point (P,Q) ?

a. y-Q=- 5/2 (x-P)
b. y-Q= 2/5 (x-P)
c. y-P=- 5/2 x-Q)
d. y-P= 2/5 (x-Q)

Answer 1

The equation of the new route, perpendicular to the old route and going through point (P,Q), is a. y - Q = -5/2(x - P).

Step-by-step explanation:

In order for the new route to be perpendicular to the old route, the slope of the new route must be the negative reciprocal of the slope of the old route. The given equation of the old route is y = (2/5)x - 4, so the slope of the old route is 2/5. Therefore, the slope of the new route would be -5/2.

Since the new route goes through point (P,Q), we can use the point-slope form of the equation to find the equation of the new route. The point-slope form is y - Q = m(x - P), where m is the slope of the new route. Plugging in the values, we get the equation y - Q = (-5/2)(x - P).

Simplifying, we have the equation y - Q = -5/2x + 5/2P.

Therefore, the correct equation of the new route is y - Q = -5/2(x - P).