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Transcript
Ellipse Conic Sections Ellipse The plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse. Ellipse - Definition An ellipse is the set of all points in a plane such that the sum of the distances from two points (foci) is a constant. d1 + d2 = a constant value. Finding An Equation Ellipse Ellipse - Equation To find the equation of an ellipse, let the center be at (0, 0). The vertices on the axes are at (a, 0), (-a, 0), (0, b) and (0, -b). The foci are at (c, 0) and (-c, 0). Ellipse - Equation According to the definition. The sum of the distances from the foci to any point on the ellipse is a constant. Ellipse - Equation The distance from the foci to the point (a, 0) is 2a. Why? Ellipse - Equation The distance from (c, 0) to (a, 0) is the same as from (-a, 0) to (-c, 0). Ellipse - Equation The distance from (-c, 0) to (a, 0) added to the distance from (-a, 0) to (-c, 0) is the same as going from (-a, 0) to (a, 0) which is a distance of 2a. Ellipse - Equation Therefore, d1 + d2 = 2a. Using the distance formula, ( x c ) 2 y 2 ( x c ) 2 y 2 2a Ellipse - Equation Simplify: ( x c ) 2 y 2 ( x c ) 2 y 2 2a ( x c ) y 2a ( x c ) y 2 2 2 2 Square both sides. ( x c ) 2 y 2 4a 2 4a ( x c ) 2 y 2 ( x c ) 2 y 2 Subtract y2 and square binomials. x 2 2 xc c 2 4a 2 4a ( x c) 2 y 2 x 2 2 xc c 2 Ellipse - Equation Simplify: x 2 2 xc c 2 4a 2 4a ( x c) 2 y 2 x 2 2 xc c 2 Solve for the term with the square root. 4 xc 4a 2 4a ( x c) 2 y 2 xc a 2 a ( x c) 2 y 2 Square both sides. xc a 2 2 a ( x c)2 y 2 2 Ellipse - Equation Simplify: xc a 2 2 a ( x c) y 2 2 2 x c 2 xca a a x 2 xc c y x 2c 2 2 xca 2 a 4 a 2 x 2 2 xca 2 a 2c 2 a 2 y 2 2 2 2 4 2 2 2 x c a a x a c a y 2 2 4 2 2 2 2 2 2 Get x terms, y terms, and other terms together. 2 2 2 2 2 2 2 2 4 x c a x a y a c a 2 Ellipse - Equation Simplify: x c 2 2 a x a y a c a 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 c a x a y a c a Divide both sides by a2(c2-a2) c 2 a2 x2 2 2 a2 c2 a2 a y 2 2 2 2 2 2 2 2 a c a a c a a c a2 x2 y2 2 1 2 2 a c a Ellipse - Equation 2 2 x y 2 1 2 2 a c a Change the sign and run the negative through the denominator. 2 2 x y 2 2 1 2 a a c At this point, let’s pause and investigate a2 – c2. Ellipse - Equation d1 + d2 must equal 2a. However, the triangle created is an isosceles triangle and d1 = d2. Therefore, d1 and d2 for the point (0, b) must both equal “a”. Ellipse - Equation This creates a right triangle with hypotenuse of length “a” and legs of length “b” and “c”. Using the pythagorean theorem, b2 + c2 = a2. Ellipse - Equation We now know….. x2 y2 2 2 1 2 a a c and b2 + c2 = a2 b2 = a2 – c2 Substituting for a2 - c2 2 2 x y 2 1 2 a b where c2 = |a2 – b2| Ellipse - Equation The equation of an ellipse centered at (0, 0) is …. 2 2 where a2 = b2 + c2 and c is the distance from the center to the foci. x y 2 1 2 a b Shifting the graph over h units and up k units, the center is at (h, k) and the equation is x h a 2 2 y k b 2 2 1 where a2 = b2 + c2 and c is the distance from the center to the foci. Ellipse - Graphing x h a 2 2 y k b 2 where a2 = b2 + c2 and c is the distance from the center to the foci. 2 1 Vertices are “a” units in the x direction an “b” units in the y direction. b a c a c b The foci are “c” units in the direction of the longer (major) axis. Graph - Example #1 Ellipse Ellipse - Graphing Ellipse - Graphing Graph: x 2 16 2 y 3 25 2 1 Graph - Example #2 Ellipse Ellipse - Graphing Graph: 2 5 x 2 y 10 x 12 y 27 0 2 Find An Equation Ellipse Ellipse – Find An Equation Find an equation of an ellipse with foci at (-1, -3) and (5, -3). The minor axis has a length of 4. Ellipse – Story Problem A semielliptical arch is to have a span of 100 feet. The height of the arch, at a distance 40 feet from the center is to be 100 feet. Find the height of the arch at its center. Ellipse – Story Problem A hall 100 feet in length is to be designed into a whispering gallery. If the foci are located 25 feet from the center, how high will the ceiling be at the center? Assignment: Wksheet #4-7**, 20-23, 33, 38, 46, 47 **Graph and find center, major vertices, minor vertices, and foci Please use graph paper!!