Standard Form of Ellipses

Algebra 2 · Axiom Academy

LESSON Standard Form of Ellipses Center it at the origin, write , and the larger denominator points straight at the major axis. 1. The Larger Denominator Marks the Major Axis Start from the standard form. Setting y=0 gives the x-intercepts ; setting x=0 gives the y-intercepts . So a is how far the ellipse reaches left and right, and b is how far it reaches up and down. Because and , the bigger a denominator is, the farther the ellipse reaches along that variable. The longer reach is the semi-major axis ; the shorter is the semi-minor axis . 2. Reading a Horizontal Ellipse ( ) Take . The denominators are 25 under x^2 and 9 under y^2 ; take a square root of each to get the two reaches. along the x-axis. Since , this is the longer reach, so the major axis is horizontal . along the y-axis — the shorter reach, so the minor axis is vertical. 3. Reading a Vertical Ellipse ( ) Now take . Here is larger than , so the longer reach is vertical — the major axis is vertical , even though the equation looks almost the same as before. b=6 is the semi-major axis, so the vertices are . a=4 is the semi-minor axis, so the co-vertices are . A point lies on the ellipse exactly when it satisfies the equation. Watch drop onto the curve: 4. Writing the Equation From the Axes Run it backward. Suppose an origin-centered ellipse reaches 6 units left and right and 10 units up and down. Square each reach to get the denominator that belongs under that variable. a=6 , so a^2=36 goes under x^2 .

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