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# Buy one, get three free!

There’s a little chance that you clicked this because of the title, which is somewhat subtle, given that we’re not marketing anything. So, if this turned out as a clickbait, we apologize straightway; but please wait, read on.

In our previous post, we talked about linear-quadratic systems. However, as that post was getting longer than we wanted, and we were gradually going off on a tangent, we decided to separate this content. Ironically, we went off on a tangent not long after we started talking about tangents. Can anything be closer to an irony?

## Tangent conditions for conic sections

The result below has been proved for the circle and the parabola ; we now consider ellipses and hyperbolas.

#### Proposition (ellipse)

(1)

(2)

has only one point of intersection, then:

1. the point of tangency has coordinates .

To prove the first statement, we utilize the fact that there’s just one point of intersection, which in turn means that the discriminant of the resulting quadratic is zero. Substitute for in equation (2):

as required. This proves the first statement. To prove the second statement, we note that the only root of the quadratic equation

is at its vertex, namely , which is the -coordinate of the tangent point. Next, we use equation (1) to get the -coordinate of the tangent point:

This verifies that the point of tangency is at .

For a circle, ; so the above result simplifies to what we got for the case of circles.

#### Proposition (unit hyperbola)

(3)

(4)

has only one point of intersection, then:

1. ;
2. the point of tangency has coordinates .

The proof is similar to the case of an ellipse, so is omitted.

## Obtaining three free tangents from one tangent

Now comes our main point, which motivated the title of this post. It has to do with circles, ellipses, and hyperbolas being generally generous, when it comes to tangents.

#### Proposition

(5)

the following statements are equivalent:

1. the line is a tangent;
2. the line is a tangent;
3. the line is a tangent;
4. the line is a tangent.

In other words, one tangent gives rise to three other tangents.

To prove the above equivalence, we first substitute for in the second equation, just like we did previously and obtain the quadratic equation

whose discriminant is . Now, if the first claim is true, then

and so .

#### Proposition

(6)

the following statements are equivalent:

1. the line is a tangent;
2. the line is a tangent;
3. the line is a tangent;
4. the line is a tangent.

The proof is just the same as in the case of a circle, so we skip it.

#### Proposition

(7)

the following statements are equivalent:

1. the line is a tangent;
2. the line is a tangent;
3. the line is a tangent;
4. the line is a tangent.

Again we omit the proof, since it is similar to the case of a circle.