Prove that y^2=4x+4 and y^2=4-4x intersect at right angles.

To find out the intersection point, we have to put the 2
equations into the relation:

4x+4 =
4-4x

For x=0, we'll have y^2 = 4*0 +
4

y^2 = 4

y1=2 and
y2=-2

To verify if the 2 curves (parabolas) are
perpendicular, we have to verify if the product of the slopes of their tangent lines, in
their intersection point, is -1.

First, let's calculate
their slopes. In order to do so, we'll calculate their
derivatives.

The derivative, with respect to x, for the
first parabola:

2y*y' = 4

y' =
4/2y

y' = 2/y, the slope of the first
parabola.

The derivative, with respect to x, for the second
parabola:

2y*y' = -4

y' =
-2/y, the slope of the second parabola.

Now, we'll check if
the product of slopes is -1.

(2/y)(-2/y) =
-4/y^2

We'll substitute y^2 by the value found at the point
of intersection, y^2=4

-4/y^2 =
-4/4=-1

The product of the slopes yields -1, so the
parabolas are intersecting eachother at right angle.