Circles: More Examples

What does it mean for a line to be tangent to a circle?

A line that is tangent to a circle is a line that touches the circle at only one point and which is, at that point, perpendicular to the radius line from the center to that same point.

This tangent-line fact, among others, is something they'll expect you to have learned in, and remembered from, a geometry class (possibly some years ago). And working with lines that are perpendicular brings in the topic of the slope of a straight line, which you'll be expected to remember from an algebra class. Et cetera.

Yeah. 🙄

• Find an equation for the circle centered at (1, −5) and tangent to the line 3x + 4y = 8.

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What is the interior / exterior of a circle?

The interior of a circle is the set of all the points inside the circle's line, having a distance from the center of less than the length of the radius. The exterior of a circle is the set of all points outside the circle's line, having distances from the center greater than the length of the radius.

• Is the point (−2, 0) in the interior or exterior of the circle with equation (x − 3)² + (y + 5)² = 49?

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At the beginning of this lesson, I'd mentioned that circles are not functions, which is part of why they can feel kind of tacked on to algebra, rather than being a proper part of algebra. You will be asked to graph circles, and you might like to check your work before you hand it in. But how, when your graphing calculator can only graphing functions?

How can my calculator graph a circle's equation?

To check your graphs, or to graph an equation to verify that it fits the requirements of an exercise, you can solve the circle equation for the two half-circle equations. Each of these half-circle equations will be a function, which you can plug into your graphing calculator.

For instance, (x − 3)² + (y + 2)² = 25 solves as follows:

Your calculator can only graph y=, not y±. But you can split the "plus-minus" into two separate equations, each of which on its own is a function. The "plus the square root of" part is the top half of the circle; the "minus the square root of" part is the bottom half. Plug these two halves into your calculator as follows:

...and graph:

The standard calculator set-up, with the tickmarks on the x-axis being wider apart than on the y-axis, will make the graph look squashed; the above was done using the "Zoom-Square" setting. Even squashed, though, your graph should suffice to confirm that the circle is centered at or near (3, −2), with a radius at or near 5.

(The gap between the upper and lower halves of the circle is an artifact of the calculator's limitations. It can't understand graphing the bits that are close to vertical.)

Make sure you memorize the center-radius form of the equation, and practice how to complete the square, how to do quick but nicely round circle graphs, and how to use your graphing calculator to check your answers. Other than involving some messy computations, circles aren't too bad — as conics go.