DEFINITION OF A ELLIPSE STANDARD FORMULAS FOR ELLIPSES Standards 4, 9, 16, 17 ELLIPSES DEFINITION OF A ELLIPSE STANDARD FORMULAS FOR ELLIPSES PROBLEM 1 PROBLEM 2 PROBLEM 3 PROBLEM 4 PROBLEM 5 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

ALGEBRA II STANDARDS THIS LESSON AIMS: Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes STANDARD 9: Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b) + c. STANDARD 16: Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it. STANDARD 17: Given a quadratic equation of the form ax + by + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation. 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

ESTÁNDAR 4: Los estudiantes factorizan polinomios representando diferencia de cuadrados, trinomios cuadrados perfectos, y la suma de diferencia de cubos. ESTÁNDAR 9: Los estudiantes demuestran y explican los efectos que tiene el cambiar coeficientes en la gráfica de funciones cuadráticas; esto es, los estudiantes determinan como la gráfica de una parábola cambia con a, b, y c variando en la ecuación y=a(x-b) + c ESTÁNDAR 16: Los estudiantes demuestran y explican cómo la geometría de la gráfica de una sección cónica (ej. Las asimptótes, focos y excentricidad) dependen de los coeficientes de la ecuación cuadrática que las representa. Estándar 17: Dada una ecuación cuadrática de la forma ax +by + cx + dy + e=0, los estudiantes pueden usar el método de completar al cuadrado para poner la ecuación en forma estándar y pueden reconocer si la gráfica es un círculo, elipse, parábola o hipérbola. Los estudiantes pueden graficar la ecuación 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Definition of Ellipse: Standards 4, 9, 16, 17 ELLIPSE Definition of Ellipse: A ellipse is the set of all points in a plane such that the sum of the distances from the foci is constant. y x d 1 + d 2 = k d 1 d 3 d 2 d 4 d 5 d 3 + d 4 = k d 6 Focus 2 Focus 1 Ellipse d 5 + d 6 = k PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARD EQUATION OF AN ELLIPSE In this case, major axis is vertical. Standards 4, 9, 16, 17 STANDARD EQUATION OF AN ELLIPSE y x (0,b) b Minor axis a Ellipse with center at (h,k) with horizontal axis has equation (-a,0) c (a,0) (x – h) (y – k) 2 = 1 a b + Major axis F 1 (-c, 0) F 2 (c, 0) In this case, major axis is horizontal. (0,-b) a = b + c 2 Major axis (0,a) y x Ellipse with center at (h,k) with vertical axis has equation F 2 (0, c) c a (x – h) (y – k) 2 = 1 a b + (-b,0) (b,0) Minor axis b In this case, major axis is vertical. F 1 (0,-c) NOTE: These two ellipses are graphed with center (0,0) a = b + c 2 a>b in both cases (0,-a) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Find the foci, a, b, and c and the equation of the ellipse below: Standards 4, 9, 16, 17 Find the foci, a, b, and c and the equation of the ellipse below: 4 2 6 -2 -4 -6 8 10 -8 -10 x y a = b + c 2 h= -4 -b -b 2 k= -1 c = a - b 2 Focus 1 = ( h + c, k) Focus 1 = ( -4+ , -1) 4 c = 25-9 2 (-4,2) Focus 1 = ( 0, -1) c = 16 2 3 Focus 2 = ( h - c, k) c= 4 Focus 2 = (-4 - , -1) 4 (1,-1) (-4,-1) 5 Focus 2 = ( -8,-1) The equation is: (x – h) (y – k) 2 = 1 a b + We can see that a=5 and b=3 (x-(-4)) (y-(-1)) 2 =1 25 9 + a = 25 2 b = 9 2 (x+4) (y+1) 2 =1 25 9 + PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Graph the following ellipse equation: Standards 4, 9, 16, 17 (x-3) (y-4) 2 =1 25 9 + Graph the following ellipse equation: (x-3) (y-4) 2 =1 25 9 + Rewriting the equation to graph it: (x-(+3)) (y-(+4)) 2 =1 25 9 + 25 > 9 So, this ellipse is horizontal. (x – h) (y – k) 2 = 1 a b + Focus 1 = ( h + c, k) h= 3 a = b + c 2 -b -b 2 Focus 1 = ( 3+ , 4) 4 k= 4 Focus 1 = ( 7, 4) a = 25 2 a = 25 2 c = a - b 2 a = 5 Focus 2 = ( h - c, k) c = 25-9 2 b = 9 2 Focus 2 = ( 3 - , 4) 4 b = 9 2 c = 16 2 Focus 2 = ( -1,4) b = 3 c= 4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Summarizing obtained information about ellipse to graph it: Standards 4, 9, 16, 17 Summarizing obtained information about ellipse to graph it: h= 3 Center (3,4) 4 2 6 -2 -4 -6 8 10 -8 -10 x y k= 4 a = 5 Major axis = 2a b = 3 Minor axis = 2b Focus 1 = ( 7, 4) Focus 2 = ( -1, 4) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

81 > 64 So, this ellipse is vertical. Standards 4, 9, 16, 17 (x+2) (y-6) 2 =1 64 81 + Graph the ellipse: (x+2) (y-6) 2 =1 64 81 + 81 > 64 So, this ellipse is vertical. (x-(-2)) (y-(+6)) 2 =1 64 81 + (x – h) (y – k) 2 = 1 b a + a = b + c 2 h= -2 -b -b 2 Focus 1 = ( h, k + c) k= 6 c = a - b 2 Focus 1 = ( -2, 6 + ) 17 a = 81 2 a = 81 2 c = 81-64 2 Focus 1 = ( -2, 6 + 4.1) a = 9 Focus 1 = ( -2, 10.1) c = 17 2 b =64 2 b = 64 2 Focus 2 = ( h, k - c) c = 17 b = 8 Focus 2 = (-2, 6 - ) 17 c 4.1 Focus 2 = ( -2, 6 – 4.1) Focus 2 = ( -2, 1.9) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Summarizing obtained information about ellipse to graph it: Standards 4, 9, 16, 17 Summarizing obtained information about ellipse to graph it: h= -2 Center (-2,6) y 4 2 6 -2 -4 -6 8 10 -8 -10 x 12 14 16 18 k= 6 a = 9 Major axis = 2a b = 8 Minor axis = 2b Focus 1 = (-2, 10.1) Focus 2 = (-2, 1.9) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standards 4, 9, 16, 17 We know that 16x + 64y -64x -384y + 384 = 0 is an ellipse. Put it in the standard form find a, b, c and the foci and graph it. (x – h) (y – k) 2 = 1 a b + 16x + 64y -64x -384y + 384 = 0 2 16x -64x + 64y -384y + 384 = 0 2 16x -16(4)x + 64y -64(6)y + 384 = 0 2 4 2 6 2 16 x - 4x + + 64 y - 6y + + 384 = 16 + 64 2 (2) 2 (3) 2 16 x - 4x + + 64 y - y + + 384 = 16 + 64 2 16 x - 4x + + 64 y - 6y + + 384 = 16 + 64 2 4 (4) 9 (9) 16(x-2) + 64(y-3) + 384 = 64 +576 2 16(x-2) + 64(y-3) + 384 = 640 2 -384 -384 16(x-2) + 64(y-3) = 256 2 256 256 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standards 4, 9, 16, 17 We know that 16x + 64y -64x -384y + 384 = 0 is an ellipse. Put it in the standard form find a, b, c and the foci and graph it. 2 (x – h) (y – k) = 1 a b + 16(x-2) + 64(y-3) = 256 2 (x-(+2)) (y-(+3)) 2 =1 16 4 + h= 2 256 256 k= 3 (x – h) (y – k) 2 = 1 a b + 16(x-2) 64(y-3) 2 =1 256 + a = 16 2 a = 16 2 a = 4 16(x-2) 64(y-3) 2 =1 256 16 64 + b = 4 2 b = 4 2 a = b + c 2 b = 2 -b -b 2 Focus 1 = ( h + c, k) c = a - b 2 (x-2) (y-3) 2 =1 16 4 + Focus 1 = ( 2+ , 3) 2 3 c = 16-4 2 Focus 1 = ( 2 + 3.5, 3) 12 2 6 2 Focus 1 = ( 5.5, 3) c = 12 2 3 3 Focus 2 = ( h - c, k) 1 c = 2 3 2 2 3 Focus 2 = ( 2 - , 3) 2 3 = 12 2 c = 2 3 2 Focus 2 = ( 2 - 3.5, 3) Focus 2 = ( -1.5, 3) c = 2 3 3.5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Summarizing obtained information about ellipse to graph it: Standards 4, 9, 16, 17 Summarizing obtained information about ellipse to graph it: h= 2 Center (2,3) 4 2 6 -2 -4 -6 8 10 -8 -10 x y k= 3 a = 4 Major axis = 2a b = 2 Minor axis = 2b Focus 1 = ( 5.5, 3) Focus 2 = ( -1.5, 3) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standards 4, 9, 16, 17 We know that 49x + 36y +392x -360y -80 = 0 is an ellipse. Put it in the standard form find a, b, c and the foci and graph it. 2 (x – h) (y – k) = 1 b a + 49x + 36y +392x - 360y - 80 = 0 2 49x +392x + 36y -360y - 80 = 0 2 49x +49(8)x + 36y -36(10)y -80 = 0 2 8 2 10 2 49 x + 8x + + 36 y -10y + -80 = 49 + 36 2 (4) 2 (5) 2 49 x + 8x + + 36 y -10 y + - 80 = 49 + 36 2 49 x + 8x + + 36 y - 10y + - 80 = 49 + 36 2 16 (16) 25 (25) 49(x+4) + 36(y-5) - 80 = 784+900 2 49(x+4) + 36(y-5) - 80 = 1684 2 +80 +80 49(x+4) + 36(y-5) = 1764 2 1764 1764 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

49 > 36 So, this ellipse is vertical. 13 Focus 2 = (-4, 5 - ) Standards 4, 9, 16, 17 We know that 16x + 64y -64x -384y + 384 = 0 is an ellipse. Put it in the standard form find a, b, c and the foci and graph it. 2 (x – h) (y – k) = 1 b a + 49(x+4) + 36(y-5) = 1764 2 (x-(-4)) (y-(+5)) 2 =1 36 49 + h= -4 1764 1764 k= 5 (x – h) (y – k) 2 = 1 b a + 49(x+4) 36(y-5) 2 =1 1764 + a = 49 2 a = 49 2 a = 7 49(x+4) 36(y-5) 2 =1 1764 49 36 + b =36 2 b = 36 2 a = b + c 2 b = 6 -b -b 2 Focus 1 = ( h, k + c) c = a - b 2 (x+4) (y-5) 2 =1 36 49 + Focus 1 = ( -4, 5 + ) 13 c = 49-36 2 Focus 1 = ( -4, 5 + 3.6) Focus 1 = ( -4, 8.6) c = 13 2 Focus 2 = ( h, k - c) c = 13 49 > 36 So, this ellipse is vertical. 13 Focus 2 = (-4, 5 - ) Focus 2 = ( -4, 5 – 3.6) c 3.6 Focus 2 = ( -4, 1.4) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Summarizing obtained information about ellipse to graph it: Standards 4, 9, 16, 17 Summarizing obtained information about ellipse to graph it: h= -4 Center (-4,5) y 4 2 6 -2 -4 -6 8 10 -8 -10 x 12 14 16 18 k= 5 a = 7 Major axis = 2a b = 6 Minor axis = 2b Focus 1 = (-4, 8.6) Focus 2 = (-4, 1.4) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved