THE EQUATION OF THE POLAR OF A POINT WITH RESPECT TO A CIRCLE (1)

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 x² + y² = 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:

x² + (mx + 4 – 4m)² = 9

x² + m²x² + 16 + 16m² + 8mx – 8m²x – 32m = 9

(m² + 1)x² + (8m – 8m²)x + (16m² – 32m + 7) = 0

(m² + 1)x² + 8m(1 – m)x + (16m² – 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 = b² – 4ac = 0

[8m(1-m)]² – 4(m² + 1)(16m² – 32m + 7) = 0

7m² – 32m + 7 = 0

By the formula to solve the quadratic equation of the form at² + 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. m₁ and m₂) 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 pole. Thus, 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 x² + y² = R² and the point K(x₁, y₁) 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:

x₁x + y₁y = R²

In the example above, x₁ = 4, y₁ = 4, and R = 3. Substituting these values ​​into the formula, we have the equation of the polar, i.e. 4x + 4y = 3², which is equivalent to 4x + 4y = 9.

Suppose that the circle is centered at (α,β) and has radius R. Let the point K(x₁,y₁) satisfy (x₁ – α)² + (y₁ – β)² ≠ R² (that is, K does not lie on the circle.) What is the equation of the polar of K? The following formula apply.

(x₁ – α)(x – α) + (y₁ – β)(y – β) = R²

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).