So if this is 1, 2, 3, 1, 2.

So the line y equals x looks like that.

Let's call this line right over here line AB.

It looks like a parabola.

Polar Line of a Point

Intersection of lines

The polar line of a point with respect to a conic.

Dot

The Dotted Line

Dots

The polar point of a line with respect to a conic.

Where is the end of this line?

let's call it y sub 1.

So let's add a straight edge here, so this is to draw a line.

Polar Point of a Line

On the left hand panel, again, we have the S shaped curve of adoption.

less than or equal.

Points and lines

is when two rays share a common vertex. and that common vertex is actually called the vertex of the angle. so A is vertex.

So that's what this equation will look like.

And that's essentially 0,0.

So the first thing we know is where the center of this hyperbola is.

Start transport from this point of the line

Start transport from this point of the curve

A point on the solution curve

If they had drawn a dotted line here that means we're not including the line.

So I have hopefully shown to you that the locus of all points that are equidistant to this point which is really just an arbitrary point.

This curve contains the point.

And let's say this parabola has a vertex at x1 comma y1.

Conic by Directrix, Focus Point

This is the locus of all points that are between the directrix, y is equal to k well, they're equidistant to the directrix, y is equal to k, and the focus a comma b.

And now we should be able to use this information to figure out the actual coordinates of the focus and the directrix of what I would call the classic parabola, y is equal to x squared.

Dash Dot

Dash Dot Dot

What's the connection?

So your distance between the two left and right points in a horizontal ellipse is the same as the distance between the two

So the points a,0, and the point minus a,0, are both on this hyperbola.

Well, the only point that sits on both lines is the intersection of those lines.

B goes straight up and down, so we can add up arbitrary multiples of b to that.

So the line will look like this.

Inversion of Point, Line or Circle

Do you see?

You go 1 2 3 minus 3, minus 1.

Let's make it 0, 3, and 4.

So there's one point that just by looking at it we could say is going to be in our locus.