Circles and Lines
We just learned how to represent a circle with an equation. We've known for a long time how to represent a line with an equation. So let's use equations to examine what happens when circles and lines interact.
We represent a generic circle with (x – h)2 + (y – k)2 = r2and a generic line with y = mx + b. Let's put a line and a circle together in the same system of equations.
(x – h)2 + (y – k)2 = r2
y = mx + b
Don't get too worried (or excited). We won't make you solve systems like this, even though you could do it. The algebra is a little tough and we're trying to focus on the geometry here. Besides, focusing on the geometry might actually help us understand the algebra.
Remember that the solution of a system of two equations like this one is the set of all points (x, y) that make both equations true at the same time. In other words, the solution of that system is the set of all points that are both on the circle and on the line. In other other words, the solution of that system is the set of points where the circle and the line intersect.
We already know that a line can intersect a circle at two, one, or zero points (it can be a secant of the circle, a tangent to the circle, or neither). Therefore, this system of equations must always have two, one, or zero solutions, depending on whether the line is a secant, a tangent, or neither.
As we said before, we won't make you solve a system like this. But we will make you do some algebra, because it's fun. And maybe we're a little sadistic.
We've talked before about cutting up a pizza, but never in Australia. Each cut made by a pizza knife stretches from one side of the pizza to the other, intersecting the crust of the pizza at two points. So each cut is a secant. Hopefully, each cut is also a diameter, so that the pizza ends up looking like this:
and not like this:
Suppose we put a coordinate grid over that pizza so you can see the coordinates of the points where the sloppy pizza man's cut intersects the edge of the pizza as well as find the equation of the circle.
Could you find the equation of the line made by the cut? Of course you could. You actually don't need to consider the circle at all: it's just finding the equation of a line given two points on the line, which is algebra-level stuff.
But let's say the pizza man is really sloppy and makes a cut that's not a secant of the pizza, but rather a tangent to the pizza. He's basically cutting the box, only touching the tiniest nub of pizza. Could you find the equation of the line then?
Let's draw a diagram of the situation. The pizza has a radius of 25 cm. Let's consider the center of the pizza to be at the origin. This sloppy pizza man, who is seriously liable to damage himself, his co-workers, or his place of employment, makes a cut tangent to the pizza at the point (7, 24).
We know one point on the line already, so let's see if we can find the slope of the line. Then we'll have enough information to write the equation of the line.
We know that the tangent line is perpendicular to the radius of the circle at the point of tangency. So if we find the slope of the radius, we can take the negative reciprocal and we have the slope of the tangent line. Ingenious!
What is the equation of the tangent line that intersects a circle centered at the origin with radius 25 at the point (7, 24)?
In this case, the radius is the line segment with endpoints (0, 0) and (7, 24). The slope of the radius is
Taking the negative reciprocal, we get a slope of for the tangent line. We now have enough to write the equation of the tangent line in point-slope form: