How to Graph Circles and Ellipses

Learn how to graph circles and ellipses step by step. Covers standard form equations, center, radius, semi-axes, completing the square, and plotting conic sections.

Circles: The Standard Form

A circle is the set of all points in a plane equidistant from a fixed center point. The standard form equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r². This equation comes directly from the distance formula: the distance from any point (x, y) on the circle to the center (h, k) is always r. To graph a circle, plot the center and then mark four key points at distance r in each cardinal direction: (h + r, k), (h - r, k), (h, k + r), and (h, k - r). Connect these with a smooth circular curve. The radius r is always the positive square root of the right-hand side of the equation.

Converting from General Form to Standard Form

Circles are sometimes given in general form: x² + y² + Dx + Ey + F = 0. To graph, convert to standard form by completing the square separately for x and y. Group the x terms and y terms, then move F to the right side. For x² + Dx, add (D/2)² to both sides; the x terms become (x + D/2)². Do the same for y. The result is (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F. The center is at (-D/2, -E/2) and the radius is the square root of the right side. If the right side is negative, the equation has no real graph. If it is zero, the graph is a single point.

Ellipses: Standard Form

An ellipse is an oval-shaped curve where the sum of distances from any point on the curve to two fixed points (the foci) is constant. The standard form of an ellipse centered at (h, k) is (x - h)²/a² + (y - k)²/b² = 1. When a > b, the ellipse is wider than it is tall: the major axis lies along the x-direction with semi-axis length a, and the minor axis is vertical with semi-axis length b. When b > a, the ellipse is taller than wide with the major axis vertical. A circle is the special case where a = b = r. The four vertices of the ellipse are at (h ± a, k) and (h, k ± b).

Graphing an Ellipse Step by Step

To graph (x - h)²/a² + (y - k)²/b² = 1, start by plotting the center (h, k). Then mark the four vertex points: move a units left and right from center along the x-axis to get (h - a, k) and (h + a, k), and move b units up and down to get (h, k + b) and (h, k - b). Draw a smooth oval curve passing through all four vertex points. The ellipse is symmetric about both its major and minor axes. For example, (x-1)²/9 + (y+2)²/4 = 1 has center (1,-2), a = 3, b = 2. Vertices: (4,-2), (-2,-2), (1,0), (1,-4). Connect with a smooth oval.

Foci of an Ellipse

The foci of an ellipse are two interior points on the major axis. Their location is determined by the relationship c² = a² - b² (when a > b) or c² = b² - a² (when b > a). The foci lie at distance c from the center along the major axis. For example, if a = 5 and b = 3, then c = √(25 - 9) = 4, so the foci are 4 units from the center along the major axis. The foci are important in the physical interpretation of ellipses: planets orbit the sun in ellipses with the sun at one focus, and elliptical reflectors direct sound or light from one focus to the other.

Eccentricity and Shape

Eccentricity e = c/a (where a is the length of the semi-major axis and c is the focal distance) describes how "stretched" an ellipse is. When e = 0, the ellipse is a circle. As e approaches 1, the ellipse becomes increasingly elongated and flat. Earth's orbit has eccentricity about 0.017 (nearly circular), while a comet's orbit can have eccentricity near 0.99 (extremely elongated). On a graph, you can visually estimate eccentricity: an ellipse that is nearly as wide as it is tall has low eccentricity, while one that is very narrow and elongated has high eccentricity.

Common Mistakes and Tips

A frequent error is confusing a² and b² when they appear on different sides of the equation, which leads to drawing the major axis in the wrong direction. Always identify which denominator is larger: the larger denominator corresponds to the major axis. Another mistake is treating a and b as the radii directly without taking square roots from the equation. Remember that if the equation is in the form ... /25 + .../9 = 1, then a = 5 and b = 3, not 25 and 9. When completing the square to convert from general form, always add the same quantity to both sides of the equation to maintain equality.

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