Geometry Equations of Circles and Parabolas Lesson (Free PDF)

Geometry math lesson: Equations of Circles and Parabolas – Maths PDF

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This circle equations lesson for Geometry covers equations of circles and parabolas (Common Core G-GPE.A.1, G-GPE.A.2), with a free PDF to print.

The graph of (x – h) squared + (y – k) squared = r squared is a circle with center (h, k) and radius r, because the equation says every point is r units from the center. Expanded equations are converted by completing the square. A parabola is the set of points equally far from a focus and a directrix.

Geometry · Common Core State Standards: G-GPE.A.1, G-GPE.A.2

Key facts in this circle equations lesson

Key facts

Circle: (x – h)^2 + (y – k)^2 = r^2, center (h, k), radius r.
Read the center with opposite signs from the equation.
Inside, on, or outside: compare (x – h)^2 + (y – k)^2 with r^2.
Complete the square to convert x^2 + y^2 + Dx + Ey + F = 0.
Parabola with focus (0, p) and directrix y = -p: x^2 = 4py.

How is a circle written as an equation?

Choose a center \((h, k)\) and a radius \(r\). A point \((x, y)\) sits on the circle exactly when its distance to the center is \(r\). Squaring the distance formula removes the root and leaves the equation below.

\[(x – h)^{2} + (y – k)^{2} = r^{2}\]

When the center is the origin, the equation shrinks to \(x^{2} + y^{2} = r^{2}\). Take \(x^{2} + y^{2} = 25\): the point \((3, 4)\) fits, since \(9 + 16 = 25\), so it lies on that circle.

Origin-centered circle of radius 5 with the point (3, 4) on it.

Circle drawn around (3, -2) with radius 5.

Where are the center and the radius in the equation?

The signs inside the parentheses are opposite to the center’s coordinates. Reading \((x + 4)^{2} + (y – 1)^{2} = 36\) gives the center \((-4, 1)\), and taking the square root of 36 gives the radius 6.

Is a given point on the circle, inside it or outside it?

Put the coordinates of the point into the left side of the equation and compare the result with \(r^{2}\): equal means the point is on the circle, smaller means inside, and larger means outside.

Point A lies on the circle, B lies outside and D lies inside.

What if the equation is not in center-radius form?

Sometimes a circle arrives expanded, as \(x^{2} + y^{2} + Dx + Ey + F = 0\). Collect the x terms together and the y terms together, add the needed squares to both sides, and factor each group to get back to the center-radius form.

Send the constant term to the right side and collect the x terms and the y terms.
Add \(\left(\frac{D}{2}\right)^{2}\) and \(\left(\frac{E}{2}\right)^{2}\) to both sides.
Write each group as a perfect square.
State the center and the radius from the result.

How do a focus and a directrix define a parabola?

Take a point called the focus and a line called the directrix. The parabola they define consists of every point whose distances to the two are equal. Place the focus at \((0, p)\) and the directrix along \(y = -p\). Setting the two distances equal and simplifying gives \(x^{2} = 4py\).

Each point of the curve is the same distance from F and from the line y = -2.

Worked example: To convert \(x^{2} + y^{2} – 6x + 4y – 12 = 0\): \((x – 3)^{2} + (y + 2)^{2} = 25\). From this form the center is \((3, -2)\) and the radius is 5.

Circle equations lesson: questions and answers

How do you write the equation of a circle?

Use (x – h)^2 + (y – k)^2 = r^2 with the center (h, k) and radius r. A circle with center (3, -2) and radius 5 has equation (x – 3)^2 + (y + 2)^2 = 25.

How do you find the center and radius from an equation?

Read h and k with opposite signs and take the square root of the right side. From (x + 4)^2 + (y – 1)^2 = 36 the center is (-4, 1) and the radius is 6.

How do you complete the square for a circle?

Group x terms and y terms, add half the coefficient squared to each group and to the other side, then factor. Both x and y need the step.

What is a parabola?

A parabola is the set of points that are the same distance from a focus and from a directrix line. For focus (0, 2) and directrix y = -2 the equation is x^2 = 8y.

How do you tell if a point is inside a circle?

Substitute it into (x – h)^2 + (y – k)^2. If the result is less than r^2, the point is inside the circle.

How to use this circle equations lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

Download the circle equations lesson as a PDF with the button below and keep it with your notes.

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