The perpendicular lines n and m are tangent to the parabola y = x2. Find the coordinates (a, b) where the two lines cross the y-axis

Question

The perpendicular lines n and m are tangent to the parabola y = x2. Find the coordinates (a, b) where the two lines cross the y-axis

Answer 1

Is this a calculus question?  I hope so, because that's what I used.  I'll outline the steps and some of my thinking:

 

• Using calculus, the derivative, which of course is the slope, would be 2x
• So whatever value we want on the right side of the parabola, the slope on the left side must be -1/(2x), so the lines are perpendicular
• The points HAVE to be symmetric, if the are to cross the y axis.  This tripped me up for a bit.  If you just play around with random numbers, you can get any two perpendicular lines that are tangent to this parabola.  For example, if x=1, then the tangent slope would be 2.  The perpendicular slope would then be -1/2.  Solving for the new (left side) x for m=-1/2 gives x=-1/4.  Those tangents are perpendicular, but they don't intersect on the y-axis.
• So...assuming they do cross the y axis, and are symmetrical, the only points that will work are the slopes of 1 and -1.
• Solving for m=1
• 1=2x
• x=1/2
• Solving for m=-1
• -1=2x
• x=-1/2
• This is important.  Our two tangent lines touch the parabola at x=1/2 and x=-1/2
• Now we need the corresponding y points, which due to symmetry are both 1/4 (x²)
• Our two lines then are
• m=1, point (1/2, 1/4)
• y-1/4=x-1/2
• m=-1, point (-1/2, 1/4)
• y-1/4=-1(x+1/2)
• y-1/4=-x-1/2
• Now let's set those two lines equal to each other
• x-1/2=-x-1/2
• 2x=0
• x=0
• Using the first equation, y-1/4=0-1/2
• y=-1/4

 

That's your point where the two lines cross the y axis, (0,-1/4)

 

Looking back, this problem seems somewhat easy to me, but it took me a while to put the pieces together.  Maybe someone will have a more straight forward solution.