Question

1.What's it means that "Find the curves that are perpendicular to the parabolas y = ax^2"?
2.how to prove that "if a line has slope m, a line perpendicular to it will have slope −1/m"?

Answer 1

1. perpendicular means the tangent lines are orthogonal at intersection point.
2. We may use vectors. (1,m) dot (1,-1/m) =0.

Answer 2

Tangent lines means each line of both curves.

Answer 3

2. if a is the angle line 1 makes with x axis, then its slope m = tan (a). Now, let's suppose the angle with the x-axis of another line is b such that tan (b) = -1/m. Your question is how come a = b-pi/2?
Remember two trig identities: tan(-x) = -tan (x) [identity 1], tan (pi/2 -x)= cot (x) = 1/tan(x) [identity 2]
so, -tan (x-pi/2) = 1/tan(x). (I applied identity 1 on identity 2]
or, tan (x-pi/2) = -1/tan(x)
Now putting a for x, we get
tan (a-pi/2) = -1/tan(a) = -1/m which we had assumed was tan (b)
hence, a-pi/2 = b

Answer 4

I believe that the first part of this question is an orthogonal trajectories problem.