Intersection of a Line and a Circle (Part 2)

In this lesson, I’ll be deriving the expression to find the length of a chord intercepted by a circle on a line. We’ll go through two methods again, one involving quadratic equations, and the other involving geometry. Let’s begin!

Method 1

Let y = mx + c be a given line, which intersects the circle x2 + y2 = a2 at two points A and B. We’re interested to find the length of AB. As discussed in the previous lesson, by solving the two equations, we’ll obtain the coordinates of the points in which the two curves intersect. And by using the distance formula on the obtained coordinates, we get the length of AB. As simple as that!

But we can skip one step and avoid the process of finding the coordinates, and still find the length. Here’s how:

Circle Length of a Chord

Let the coordinates of the points be A(x1, y1) and B(x2, y2). Now AB = \( \sqrt{(x_1-x_2)^2 +(y_1-y_2)^2} \). Now since the points A and B lie on the given line, the slope of AB must be equal to the slope of the line, i.e. m.

That means, (y1 – y2)/(x1 – x2) = m, or (y1 – y2) = m(x1 – x2). We’ll now substitute this value into the expression for AB, to obtain AB = |x1 – x2|\( \sqrt{1+m^2} \). Looks a bit better.

Now what about the term |x1 – x2|? Do we need to find the roots x1 and x2? We don’t. We’re only interested in their difference.

Let’s come back to the quadratic which we formed previously: (1 + m2)x2 + 2cmx + c2 – a2­ = 0.

The difference of the roots is given by the expression |x1 – x2| = \( \sqrt{(x_1+x_2)^2 -4x_1x_2} \)

The value of the sum of the roots, i.e. x1 + x2 will be – 2cm/(1 + m2) and the product x1x2 will be (c2 – a2)/(1 + m2). Plug in the values in the above expression, and you’re done. I won’t write the final expression, as it is quite complicated and you don’t have to memorize it anyways.

Method 2

We have an alternate simpler method based on geometrical properties of the circle. If we’re able to find the distance of the chord from the center of the circle (say, ‘d’) then the required length will be 2\(\sqrt{r^2-d^2}\), where r is the radius of the circle. Can you see it?

Circle Length of a Chord

And how will we find ‘d’? Simple. By find the distance of the line from the center of the circle, which in this case will be given by, d = |c|/ \( \sqrt{1+m^2} \)

This method will be preferred in finding the chord’s length, but won’t be applicable in the case of the other conic sections, where we’ll have to use the previous method.

And that’ll be all for the current lesson.

Lesson Summary

  1. The length of the chord cut by a line on any conic section is given by |x1 – x2|\( \sqrt{1+m^2} \)
    where m is the slope of the given line, and |x­1 – x2| is the difference of the roots of the quadratic equation, obtained by solving the line and the curve.
  2. In case of the circle the length can also be calculated using the expression 2\(\sqrt{r^2-d^2}\)
    where r is the radius of the circle, and d is the distance of the line from the center of the circle.

The next lesson will cover a few examples to illustrate this concept.

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