# Question #60d5b

##### 1 Answer

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.

#### Explanation:

**STEP 1:**

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,

y =

So, the points of tangency are [2, 2

Now, each tangent line (y = mx + b) and the parabola

y = mx + b = m[

Now, simplify ...

m

This is a quadratic equation, so use the quadratic formula to solve for y:

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

2

-2

Now that we know the slopes of the "tangent" lines, we can calculate the slope of the "normal" lines.

**STEP 2:**

The slope of the normal line ( m' ) is always the negative reciprocal of the slope of the tangent line (m).

If m =

If m =

**STEP 3:**

Calculate the two normal lines at the points of tangency:

[2, 2

y = m'x + b

y =

The second normal line is:

y =

Hope that helped!