This post introduces the concept of polars. There are several approaches to understanding the concept. The approach adopted in this article is through the points of tangency on a circle. [Regarding the notion of the points of tangency, the readers may refer to the articles The Equations of The Tangent Lines to A Circle (1) or The Equations of The Tangent Lines to A Circle (2).]

**Example 1**

Given the circle x^{2} + y^{2} = 9, what are the equations of the tangent lines that pass through the point K(4,4)?

**Answer**

Let the tangent lines have equations of the form y = mx + c. Since the lines pass through K(4,4), the following must hold: 4 = 4m + c. From this, we have c = 4 – 4m. So, the tangent lines have equations of the form: y = mx + (4 – 4m). Substituting this into the equation of the circle, we get:

x^{2} + (mx + 4 – 4m)^{2} = 9

x^{2} + m^{2}x^{2} + 16 + 16m^{2} + 8mx – 8m^{2}x – 32m = 9

(m^{2} + 1)x^{2} + (8m – 8m^{2})x + (16m^{2} – 32m + 7) = 0

(m^{2} + 1)x^{2} + 8m(1 – m)x + (16m^{2} – 32m + 7) = 0 …………………………………………………………………………………………………………… (*)

Note that (*) is a quadratic equation in x.

Remember that the tangent line must intersect the circle at one and only one point. Mathematically speaking, it is equivalent to saying that (*) has a double root. Thus, the discriminant (D) of the quadratic equation must be zero! It follows that:

D = b^{2} – 4ac = 0

[8m(1-m)]^{2} – 4(m^{2} + 1)(16m^{2} – 32m + 7) = 0

7m^{2} – 32m + 7 = 0

By the formula to solve the quadratic equation of the form at^{2} + bt + c = 0 with a ≠ 0, we obtain two possible values of *m*, namely:

Since there are two possible values of *m*, there are two tangent lines, namely and , which pass through K(4,4). To determine the equations of the lines, substitute each m (i.e. *m*_{1} and *m*_{2}) into the line equation y = mx + (4 – 4m). This yields the following equations.

See the figure below.

In the figure, the points A and B are the points of intersection of each line and the circle. In fact, they are the points of tangency, where lines and touch the circle. The line that passes through A and B, namely g, is called a * polar*. More specifically,

**g is the polar of the point K**. K itself is called a

*. Thus,*

**pole**

**g is the polar of the pole****K**. What is the equation of the polar?

Having known the equations of the tangent lines, we can obtain the coordinates of the points A and B above. After the coordinates are obtained, the equation of the line that passes through A and B can be determined. The details of this are quite complicated (please try). However, the equation of the polar is **4x + 4y = 9**. Is there any shortcut to get the result? Fortunately, there is!

Given the circle having the equation x^{2} + y^{2} = R^{2} and the point K(x_{1}, y_{1}) with (K does not lie on the circle), the equation of the polar of K with respect to the circle can be determined as follows:

* x*_{1}*x* + *y*_{1}*y* = R^{2}

In the example above, x_{1} = 4, y_{1} = 4, and R = 3. Substituting these values into the formula, we have the equation of the polar, i.e. 4x + 4y = 3^{2}, which is equivalent to **4x + 4y = 9**.

Suppose that the circle is centered at (α,β) and has radius R. Let the point K(x_{1},y_{1}) satisfy (x_{1} – α)^{2} + (y_{1} – β)^{2} ≠ R^{2 }(that is, K does not lie on the circle.) What is the equation of the polar of K? The following formula apply.

**(x _{1} – α)(x – α) + (y_{1} – β)(y – β) = R^{2}**

For a further discussion of the formula, we refer the readers to The Equation of The Polar of A Point with Respect to A Circle (2).