Ellipse Pre-Calculus Notes PDF

ELLIPSE PRE-CALCULUS WEEK 5: 1ST QUARTER LEARNING OBJECTIVES: Define an ellipse Determine the parts of Ellipse Determine the standard form of equation of an ellipse PARTS OF ELLIPSE  It is the midpoint inside of the CENTER (C) ellipse curve and denoted by the...

ELLIPSE PRE-CALCULUS WEEK 5: 1ST QUARTER LEARNING OBJECTIVES: Define an ellipse Determine the parts of Ellipse Determine the standard form of equation of an ellipse PARTS OF ELLIPSE  It is the midpoint inside of the CENTER (C) ellipse curve and denoted by the symbol C.  - are the main points of the curve VERTEX (V) expressed by the symbols v.  - are the fixed points of FOCI (f) the Ellipse and it is the plural form of focus. - this point is located between the center and the vertices. - they are represented by the symbols 𝐹1 and 𝐹2  also called directrices. DIRECTRIX LINES (D.L.) these are the lines which are located outside of the ellipse. MINOR AXIS - it divides the curve into two symmetrical parts. - the end points of this axis are called intercepts. - the length of the minor axis is 2b  - always longer in MAJOR AXIS/ PRINCIPAL AXIS length compare to minor axis - the endpoints of the Principal or major axis are the vertices - the length of this axis is 2a  - is the distance between one of FOCAL DISTANCE the focus to the Center. - it is denoted by the symbol c  - it is the plural form of the LATERA RECTA latus rectum - the length of the lateral 2𝑏2 rectum is 𝑎  - is the ratio of the distances from ECCENTRICITY (E) the fixed point to the point P and from point P to the directrix line. - the value of the eccentricity can 𝒄 be obtained from the formula of 𝒂 Standard equation of an ellipse Standard equation of an ellipse - If the center is at the origin, then the equation takes on two forms Major axis is horizontal Major axis is vertical 2 2 2 2 𝑥 𝑦 𝑥 𝑦 2 + 2 = 1 2 + 2 =1 𝑎 𝑏 𝑏 𝑎 Examples 𝑥 2 𝑦 2 + =1 25 9 𝑉𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ℎ ± 𝑎, 𝑘 𝐶𝑜 − 𝑉𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ℎ, 𝑘 ± 𝑏 𝐹𝑜𝑐𝑖: ℎ ± 𝑐, 𝑘 𝑏2 𝐿𝑅 1&2: ℎ + 𝑐, 𝑘 ± 𝑎 𝑏2 𝐿𝑅 3&4: ℎ − 𝑐, 𝑘 ± 𝑎 Examples 𝑥 2 𝑦 2 2𝑏 2 18 + =1 𝐿. 𝑅. = = = 3.6 25 9 𝑎 4 𝑏2 9 Focus to endpoint of L.R. = = = 1.8 Principal Axis: Horizontal 𝑉1 = 5, 0 𝑎 𝑏2 4 Center: (0, 0) 𝑉2 = −5, 0 𝐿. 𝑅. 1&2 = (ℎ + 𝑐, 𝑘 ± ) 𝑐 𝐿. 𝑅. 1 4, 1.8 𝑎2 = 25 𝐶𝑉1 = 0, 3 𝐿. 𝑅. 2 4, −1.8 𝑐= − 𝑎2 𝑏2 2 𝑎=5 𝐶𝑉2 = 0, −3 𝑏 𝑐 = 25 − 9 𝐿. 𝑅. 3&4 = (ℎ − 𝑐, 𝑘 ± ) 𝑐 = 16 𝑐 𝑏2 = 9 𝐹1 = 4, 0 𝐿. 𝑅. 3 −4, 1.8 𝑏=3 𝑐=4 𝐹2 = −4, 0 𝐿. 𝑅. 4 −4, −1.8 Examples 𝑥 2 𝑦 2 + =1 9 36 𝑉𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ℎ, 𝑘 ± 𝑎 𝐶𝑜 − 𝑉𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ℎ ± 𝑏, 𝑘 𝐹𝑜𝑐𝑖: ℎ, 𝑘 ± 𝑐 𝑏2 𝐿𝑅 1&2: ℎ ± , 𝑘 + 𝑐 𝑎 𝑏2 𝐿𝑅 3&4: ℎ ± , 𝑘 − 𝑐 𝑎 Examples 𝑥 2 𝑦 2 2𝑏 2 18 𝐿. 𝑅. = = =3 + =1 𝑎 6 9 36 𝑏2 9 Focus to endpoint of L.R. = = = 1.5 Principal Axis: Vertical 𝑉1 = 0, 6 𝑎 6 𝑏2 Center: (0, 0) 𝑉2 = 0, −6 𝐿. 𝑅. 1&2 = (± 𝑐 , 𝑐) 𝐿. 𝑅. 1 1.5, 5.196 𝑎2 = 36 𝐶𝑉1 = 3, 0 𝐿. 𝑅. 2 −1.5, 5.196 𝑐= − 𝑎2 𝑏2 2 𝑎=6 𝐶𝑉2 = −3, 0 𝑏 𝑐 = 36 − 9 𝐿. 𝑅. 3&4 = (± , −𝑐) 𝑐 = 27 𝑐 𝑏2 = 9 𝐹1 = 0, 5.196 𝐿. 𝑅. 3 1.5, −5.196 𝑏=3 𝑐 = 5.196 𝐹2 = 0, −5.196 𝐿. 𝑅. 4 −1.5, −5.196 Examples (𝑥 −5)2 (𝑦 −4)2 + =1 49 25 𝑉𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ℎ ± 𝑎, 𝑘 𝐶𝑜 − 𝑉𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ℎ, 𝑘 ± 𝑏 𝐹𝑜𝑐𝑖: ℎ ± 𝑐, 𝑘 𝑏2 𝐿𝑅 1&2: ℎ + 𝑐, 𝑘 ± 𝑎 𝑏2 𝐿𝑅 3&4: ℎ − 𝑐, 𝑘 ± 𝑎 Examples (𝑥 − 5)2 (𝑦 − 4)2 2𝑏 2 50 𝐿. 𝑅. = = = 7.14 + =1 𝑎 7 49 25 𝑏2 25 Focus to endpoint of L.R. = = = 3.57 Principal Axis: Horizontal 𝑉1 = 12, 4 𝑎 7 𝑏2 Center: (5, 4) 𝑉2 = −2, 4 𝐿. 𝑅. 1&2 = ℎ + 𝑐, 𝑘 ± 𝑐 𝐿. 𝑅. 1 = 9.90, 7.57 𝑎2 = 49 𝐶𝑉1 = 5, 9 𝐿. 𝑅. 2 = 9.90, 0.43 𝑐= 𝑎2 − 𝑏2 𝑏2 𝑎=7 𝐶𝑉2 = 5, −1 𝐿. 𝑅. 3&4 = ℎ − 𝑐, 𝑘 ± 𝑐 = 49 − 25 𝑐 2 𝑐 = 24 𝐿. 𝑅. 3 0.10, 7.57 𝑏 = 25 𝐹1 = 9.90, 4 𝑐 = 4.90 𝐿. 𝑅. 4 0.10, 0.43 𝑏=5 𝐹2 = 0.10, 4 Examples 2 2 (𝑥 + 3) 𝑦 + =1 36 64 𝑉𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ℎ, 𝑘 ± 𝑎 𝐶𝑜 − 𝑉𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ℎ ± 𝑏, 𝑘 𝐹𝑜𝑐𝑖: ℎ, 𝑘 ± 𝑐 𝑏2 𝐿𝑅 1&2: ℎ ± , 𝑘 + 𝑐 𝑎 𝑏2 𝐿𝑅 3&4: ℎ ± , 𝑘 − 𝑐 𝑎 Examples (𝑥 + 3) 2 𝑦 2 2𝑏 2 72 𝐿. 𝑅. = = =9 + =1 𝑎 8 36 64 𝑏2 36 Focus to endpoint of L.R. = = = 4.5 Principal Axis: Vertical 𝑉1 = −3, 8 𝑎 8 𝑏2 Center: (-3, 0) 𝑉2 = −3, −8 𝐿. 𝑅. 1&2 = ℎ ± 𝑐 ,𝑘 +𝑐 𝐿. 𝑅. 1 = −7.5, 5.29 𝑎2 = 64 𝐶𝑉1 = −9, 0 𝐿. 𝑅. 2 = 1.5 5.29 𝑐= 𝑎2 − 𝑏2 𝑏2 𝑎=8 𝐶𝑉2 = 3, 0 𝐿. 𝑅. 3&4 = ℎ ± ,𝑘 −𝑐 𝑐 = 64 − 36 𝑐 2 𝑐 = 28 𝐿. 𝑅. 3 −7.5, −5.29 𝑏 = 36 𝐹1 = −3, 5.29 𝑐 = 5.29 𝐿. 𝑅. 4 1.5, −5.29 𝑏=6 𝐹2 = −3, −5.29 General to Standard 16𝑥 2 + 4𝑦 2 − 32𝑥 − 16𝑦 − 32 = 0 Standard to general 16𝑥 2 + 4𝑦 2 − 32𝑥 − 16𝑦 − 32 = 0 Standard to General 2 2 (𝑥 + 1) (𝑦 − 1) + =1 25 4 General to Standard 2 2 (𝑥 + 1) (𝑦 − 1) + =1 25 4 Thank you