1003 10.2 Ellipses The point B10, b2 is on the ellipse in Figure 15, so we have the following. d1B, F2 + d1B, F′2 = 2a 21-c22 + b2 + 2c2 + b2 = 2a Substitute. 22c2 + b2 = 2a Combine like terms. 2c2 + b2 = a Divide by 2. c2 + b2 = a2 Square each side. b2 = a2 - c2 Subtract c2. Replacing a2 - c2 with b2 in equation 112 gives the standard form of the equation of an ellipse centered at the origin with foci on the x-axis. x2 a2 + y2 b2 =1 If the vertices and foci were on the y-axis, an almost identical derivation could be used to obtain the following standard form. x2 b2 + y2 a2 =1 Standard Forms of Equations for Ellipses The ellipse with center at the origin and equation x2 a2 + y2 b2 =1 1where a 7b2 has vertices 1{a, 02, endpoints of the minor axis 10, {b2, and foci 1{c, 02, where c2 = a2 - b2. The ellipse with center at the origin and equation x2 b2 + y2 a2 =1 1where a 7b2 has vertices 10, {a2, endpoints of the minor axis 1{b, 02, and foci 10, {c2, where c2 = a2 - b2. a 0 c –a –c –b b Major axis on x-axis x y 0 b –b –a –c a c Major axis on y-axis x y Do not be confused by the two standard forms. • In the first form, a2 is associated with x2. • In the second form, a2 is associated with y2. In practice it is necessary only to find the intercepts of the graph—if the x-intercepts are farther from the center of the ellipse than the y-intercepts, then the major axis is horizontal; otherwise, it is vertical. When using the relationship c2 =a2 −b2, choose a2 and b2 so that a2 +b2. LOOKING AHEAD TO CALCULUS Methods of calculus can be used to solve problems involving ellipses. For example, differentiation is used to find the slope of the tangent line at a point on the ellipse, and integration is used to find the length of any arc of the ellipse.
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