Normal line.

(b) Nazim Jasim

Extreme normal line.

Find the equation of the extreme normal line.

This is the equation of the normal line.

So if we want the equation of the normal line at this extreme point, right here, the one that creates the extreme normal

So this is a normal line, right there.

The extreme normal line is shown as a thick

What is the equation of the normal line there?

So that's my equation of the normal line.

And they're saying that a normal line is something, when the first quadrant intersection with the parabola is normal to the parabola.

line intersected here, in the second quadrant.

Now, the normal line slope is perpendicular to this.

Once the normal line passes the extreme normal line, the x coordinates of their second quadrant intersections what the parabola start to increase.

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5 normal lines are shown in the figure.

This is the extreme normal line, right there.

So we could say that the equation of the normal line,

So the intersection of our normal line and our parabola is equal to this.

So this point right here will be on the normal line.

So this normal line will also have this point.

We're just figuring out where the normal line and the parabola intersect.

We've just figured out the x0 value that gives us our extreme normal line.

This value right here, that gives us the extreme normal

Now, they want us to figure out the equation of the extreme normal line.

Well, the equation of the extreme normal line we already figured out right here.

The equation of the normal line is that thing, right there.

So this is the equation of the normal line, in mx plus b form.

So the thread of discovery that we have really is a thread.

So these are the two points at which the normal curve and our parabola intersect.

The extreme normal line is when our second quadrant intersection essentially achieves a maximum point.

Or even more important, what happened to the intersection of the normal line in the second quadrant?

And so we get, the point at which they intersect, our normal line and our parabola, and this is quite a hairy problem.

So if I were to draw a tangent line right there, this line is normal to that tangent line.

We care about the normal line when we're in the first quadrant, we're in all of these values right there.

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So the slope of normal line at x0 will be the negative inverse of this, because this is the slope of the tangent line x0.

Now, what is the equation of the normal line at x0 let's say that this is my x0 in question.

You say, y minus the y point, which is just x0 squared, that's that right there, is equal to the slope of the normal line minus 1 over 2 x0 times x minus the x point that we're at.

So we just add 1 2 to both sides of this equation, and we get our extreme normal line equation, which is y is equal to minus square root of 2 over 2x.

So you can kind of view this as the highest value, or the smallest absolute value, at which the normal line can intersect in the second quadrant

So the equation of our normal line, assuming I haven't made any careless mistakes, is equal to, so y minus 1 2 is equal to, let's see.

Let us define a normal line as a line whose first quadrant intersection with the parabola is perpendicular to the parabola.

The 2 normal lines of a pair have the same second quadrant intersection with the parabola, but 1 is above the extreme normal line, in the first quadrant, the other is below it.

And then, if you go even smaller x values in the first quadrant then your normal line starts intersecting in the second quadrant, further and further negative numbers.