Find the tangent line to the parabola at the point x = 2, then find the normal line (perpendicular) to that tangent line at the point x = 2.
Find the tangent line(s) to the parabola at x = 2. If you graph this parabola, you will see that there are actually TWO tangent lines at the point x = 2. So, there are two normal lines ... one for each tangent line.
At x = 2,
So, the points of tangency are [2, 2
Now, each tangent line (y = mx + b) and the parabola
y = mx + b = m[
Now, simplify ...
This is a quadratic equation, so use the quadratic formula to solve for y:
Now, the discriminant (the part in the square root) must equal zero because there is only one point where the tangent line and the parabola intersect. So, this whole mess simplifies to:
y = 2/m
Finally, we can solve for m, the slope of the tangent lines because we know the values of y at the point of tangencies: (2
Now that we know the slopes of the "tangent" lines, we can calculate the slope of the "normal" lines.
The slope of the normal line ( m' ) is always the negative reciprocal of the slope of the tangent line (m).
If m =
If m =
Calculate the two normal lines at the points of tangency:
y = m'x + b
The second normal line is:
Hope that helped!