VIEWS: 20 PAGES: 4 POSTED ON: 2/26/2010 Public Domain
Harford Community College Summer 2008 MA101 College Algebra Section 5.2 Ellipses An ellipse is a conic section formed when a plane intersects a right circular cone. If the angle of intersection is perpendicular to the axis of the cone, then the ellipse is a special form known as a circle. Otherwise, an ellipse is formed. An ellipse is the set of all points in the plane the sum of whose distance from two points (not on the ellipse) is a positive constant. In other words, an ellipse has two foci (F1 and F2); any point in the plane whose distance from F1 plus the distance from F2 is equal to some constant is on that ellipse. Change the foci or the fixed constant and you change the ellipse. An ellipse has two axes of symmetry; the major axis is the longer axis; the minor axis is the shorter axis. The points where the major axis intersect the ellipse are the vertices. Both foci lie on the major axis. The center of the ellipse is the point at which the major and minor axes intersect. The semi major axis extends from the center of the ellipse to the vertex; the semi-minor axis extends from the center to the intersection with the ellipse along the minor axis. Standard form The standard form of an ellipse is given by x h y k 1, a b 2 2 a2 b2 x h y k 2 2 1, a b b2 a2 Where (h, k) is the center of the ellipse.The length of the major axis is 2a; the length of the minor axis is 2b. The upper definition is for an ellipse with major axis parallel to the horizontal. Notice that x is in the numerator of the expression with a in the denominator. The vertices are (h + a, k) and (h – a, k). The foci are (h + c, k) and (h – c, k), where c2 = a2 – b2. Similarly, the lower definition is for an ellipse with major axis parallel to the vertical. Notice that y is in the numerator of the expression with a in the denominator. The vertices are (h, k + a) and (h, k – a). The foci are (h, k + c) and (h, k – c), where c2 = a2 – b2 Example: Find the center, foci, vertices major and minor axis for an ellipse defined by Harford Community College Summer 2008 MA101 College Algebra Section 5.2 Ellipses 9 x 2 16 y 2 36 x 16 y 104 0 9 x 2 36 x 16 y 2 16 y 104 9 x 2 4 x 16 y 2 y 104 1 9 x 2 4 x 4 16 y 2 y 104 36 4 4 2 1 9 x 2 16 y 144 2 2 2 1 16 y 9 x 2 2 2 144 144 144 144 2 1 x 2 2 y 2 1 16 9 2 1 y x 2 2 1 2 42 32 The center is (-2, ½). The length of the major axis is 8, the length of the minor axis is 6. The major axis is parallel to the x axis (horizontal). The vertices are (2, ½) and (-6, ½). To find the foci solve c2 a 2 b2 c 2 16 9 7 c 7 1 1 The foci are 2 7, and 2 7, 2 2 The process is: 1) Rearrange terms, moving the constant(s) to the right side. 2) Factor 3) Complete the square 4) Divide by the right side Example: Find the equation of an ellipse whose center is (-4, 1) and whose minor axis is parallel to the vertical and of length 8. The ellipse passes through the point (0, 4). Since the minor axis is parallel to the vertical, the major axis is parallel to the x-axis and the form of the ellipse will be x 4 y 1 2 2 1 a2 42 Since (0, 4) is on the ellipse, it must satisfy this equation. Substituting, Harford Community College Summer 2008 MA101 College Algebra Section 5.2 Ellipses 0 4 4 1 2 2 1 a2 42 16 9 1 a 2 16 16 7 a 2 16 7 16 a 2 16 256 a2 7 256 16 16 7 a 7 7 7 So the standard form is x 4 y 1 2 2 2 1 16 7 42 7 Eccentricity The eccentricity of an ellipse is defined as the ratio of the distance from the center to the foci to the distance from the center to the vertices. c e . As c → 0, the ellipse collapses into a circle. As c → a, the ellipse becomes a thinner and more elongated. Example: Find the equation in standard form of an ellipse with Foci at (0, -3) and (0, 3), with eccentricity of ¼ . We know that c = 3, so we can solve the eccentricity for a. 1 3 4 a a 12 We also know that c2 = a2 – b2 so 9 12 b 2 b2 3 b 3 We know that the midpoint is the origin from the midpoint formula. The major axis lies on the y axis, so the form will be similar to x h y k 2 2 1, a b b2 a2 Harford Community College Summer 2008 MA101 College Algebra Section 5.2 Ellipses The result of all this knowledge is: x 0 y 0 2 2 1 3 2 122 x2 y2 1 3 2 122 Homework: Section 5.2 Exercises pp 492 – 495 #21, 27, 28, 43, 51