Treating this as a calculus problem:
The parabola is symmetric with regard to the y-axis, so what we say about the right side is reflected on the left.
y = k - x^2
y' = -2x
We want the slope to be -1 on the right (45° downward) so it's 1 on the left (45° upward), so they meet at a right angle:
-1 = -2x
x = 1/2
and similarly x = -1/2 on the left.
From these x values to x=0, these tangents go 1/2 unit higher due to the unitary nature of the slopes. At the points of tangency, the y value is k - 1/4, and so at their intersection the y-value is k + 1/4.
k + 1/4 = 1
k = 3/4.
Posted by Charlie
on 2011-10-28 13:12:55