Question

find the points of intersection of

(a) the tangents to the graph of y=2+2x-x^2 at the points where the graph meets the x-axis.

(b) the normals to the graph at these points.

(c) find the area of the triangle created from the two intersections and the point (0,0)

Answer 1

Find the roots of the polynomial.

Then find the slope at those points. y'=2-2x

Then use the point-slope form to get the line equations.

Do the same for the normals, with slope -1/y'

Find the intersections, and then you can measure the triangle.

Here's a start, showing the tangents:

http://www.wolframalpha.com/input/?i=plot+y%3D2%2B2x-x%5E2,y%3D2%E2%88%9A3+(x-1%2B%E2%88%9A3),+y%3D-2%E2%88%9A3+(x-1-%E2%88%9A3)