The Circle

Try this

Drag only $F$ . Does the center move? Which quantity changes?
Drag only $D$ . Which way does the center go? Each time $D$ changes by $2$ , how far does the center move?
Drag $F$ upward and keep an eye on the reading $D^{2} + E^{2} - 4 F$ . When it reaches $0$ , what is left in the picture? And a little beyond that?
Drag the point $P$ . When it is inside, on, and outside the circle, what sign does the reading “Left side at $P$ ” have?

The equation above has three features: it is a quadratic equation in $x, y$ ; $x^{2}$ and $y^{2}$ have the same coefficient; and there is no $x y$ -term. Conversely, every equation with these three features (once the square-term coefficients are made $1$ ) can be completed to

(x + \frac{D}{2})^{2} + (y + \frac{E}{2})^{2} = \frac{D ^{2} + E ^{2} - 4 F}{4}

The left side is the square of the distance from the point $(x, y)$ to $(- \frac{D}{2}, - \frac{E}{2})$ , so everything depends on the right side.

Predict first

The equation $x^{2} + y^{2} - 6 x + 4 y + k = 0$ represents a circle. What is the range of $k$ ?

k < 13

k < 52

k > 13

k \leq 13

Choose an answer first; then you can see the result.

Why

Completing the square translates the “general form” into “distance to a point”. Work through the example below and the figure follows along.

Completing the square: x² + y² + 6x − 8y = 0

Step 1 / 6

Group by letter
Put the $x$ terms together and the $y$ terms together: $(x^{2} + 6 x) + (y^{2} - 8 y) = 0$ . The two groups do not interfere, so you can complete each square separately.

$(x^{2} + 6 x) + (y^{2} - 8 y) = 0$
Complete the square in x
Complete the square in y
Move the constant to the right
In general
Substitute a point

The last step is very useful: the constant term $F$ is the value at the origin. If $F < 0$ the origin is inside the circle, if $F = 0$ the circle passes through the origin, and if $F > 0$ the origin is outside.

The tangent at a point of the circle

The tangent to the circle $x^{2} + y^{2} = r^{2}$ at a point $(x_{0}, y_{0})$ on it is perpendicular to the radius, whose direction is $(x_{0}, y_{0})$ . So the tangent is

x_{0} x + y_{0} y = r^{2} .

How to remember it: write $x^{2}$ as $x_{0} x$ and $y^{2}$ as $y_{0} y$ . This “replacement rule” comes back in stop 13 for the ellipse, the hyperbola and the parabola.

Practice

Exercise 1Basic

Find the radius of the circle $x^{2} + y^{2} + 4 x - 10 y + 20 = 0$ .

Exercise 2Basic

The graph of $x^{2} + y^{2} + 2 x - 4 y + m = 0$ is a single point. Find $m$ .

Exercise 3Advanced

A circle passes through the three points $O (0, 0)$ , $A (4, 0)$ and $B (0, 6)$ . Give the coordinates of its center (in the form $(a, b)$ ).

Exercise 4Challenge

From the point $P (7, 1)$ draw a tangent to the circle $x^{2} + y^{2} - 4 x + 2 y - 4 = 0$ , touching it at $T$ . Find $∣ P T ∣^{2}$ .

Try this

Why

Completing the square: x² + y² + 6x − 8y = 0

The tangent at a point of the circle

Practice

Mistake Museum · Exhibits from this stop

Seeing $x^2+y^2$ and calling it a circle

Reading off the center when the square terms do not have coefficient 1