Normal line to a parabola and its other point of intersection: Find the normal line to at the point . Then determine the other point, , at which it intersects the curve.

We can find the tangent line at . Then the normal line will be perpendicular to the tangent line.

The derivative, evaluated at , will give us , the slope of the tangent line:

. At , we have .

Because they are perpendicular, the slope of the normal line is

Now we can write the equation of the normal line in slope-intercept form

.

Since this line goes through the given point, we can use the coordinates to determine :

, or .

So the equation of the normal line is .

To find the other point of intersection of this line and the original curve, we evaluate both at the unknown coordinate :

Solving for :

, or .

Using the quadratic formula, we find that or .

Since is the given point, we see that the other point of intersection must be at .

The corresponding y-coordinate is found from the equation of the original curve: