For what value of k is the parabola y=k-x^2 viewed at a right angle * from point A(0,1)?


* i.e. displaying a right angle between the two tangents.

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