Conic Sections: Equations and Types PDF

𝑨𝒙 + 𝑩𝒚 + 𝑪 = 𝟎 𝑨𝒙𝟐 + 𝑩𝒙𝒚 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎   𝟐 𝟐 𝑨𝒙 + 𝑩𝒙𝒚 + 𝑪𝒚 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎  2 2 𝑥 + 𝑦 + 3𝑥 + 4 = 0 𝐴𝑥 2 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 𝐵 − 4𝐴𝐶 A =1...

𝑨𝒙 + 𝑩𝒚 + 𝑪 = 𝟎 𝑨𝒙𝟐 + 𝑩𝒙𝒚 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎   𝟐 𝟐 𝑨𝒙 + 𝑩𝒙𝒚 + 𝑪𝒚 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎  2 2 𝑥 + 𝑦 + 3𝑥 + 4 = 0 𝐴𝑥 2 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 𝐵 − 4𝐴𝐶 A =1 2 = 0 −4(1)(1) B =0 = −4 C =1 CIRCLE By Using Conic Discriminant:  2 𝑥 + 10𝑥 + 4 = 0 𝐴𝑥 2 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 𝐵 − 4𝐴𝐶 A =1 2 = 0 −4(1)(0) B =0 =0 C =0 PARABOLA By Using Conic Discriminant:  2 2 4𝑥 + 4𝑦 − 20𝑥 − 32𝑦 + 81 = 0 By Using Conic Discriminant:  2 2 25𝑥 + 𝑦 − 100𝑥 − 125 = 0 By Using Conic Discriminant:  2 2 9𝑥 − 𝑦 − 54𝑥 − 8𝑦 − 59 = 0 By Using Conic Discriminant:  2 𝑦 − 72𝑥 − 153 = 0 By Using Conic Discriminant:  2 2 𝐴𝑥 + 𝐵𝑥𝑦 + 𝐶𝑦 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 ≠ ≠  2 2 𝑥 + 𝑦 + 4𝑥 − 6𝑦 − 12 = 0 2 2 25𝑥 + 4𝑦 − 50𝑥 + 8𝑦 − 71 = 0 By Observing the Quadratic Terms  2 2 9𝑥 − 4𝑦 − 18𝑥 − 16𝑦 − 43 = 0 2 𝑥 + 8𝑥 + 16𝑦 + 64 = 0 By Observing the Quadratic Terms  2 2 16𝑥 + 81𝑦 − 324𝑦 − 972 = 0 2 2 16𝑥 − 4𝑦 + 24𝑦 − 100 = 0 By Observing the Quadratic Terms  2 10𝑦 + 57 = −𝑦 + 8𝑥 𝐵𝑦 𝐶𝑜𝑛𝑖𝑐 𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑇𝑒𝑟𝑚𝑠  2 2 3𝑥 − 5𝑥 + 54 = 3(5 − 3𝑦 ) 𝐵𝑦 𝐶𝑜𝑛𝑖𝑐 𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑇𝑒𝑟𝑚𝑠  𝑩𝟐 − 𝟒𝑨𝑪 𝑸𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 𝑻𝒆𝒓𝒎 𝑻𝒚𝒑𝒆 𝒐𝒇 𝑪𝒐𝒏𝒊𝒄 𝑥 + 5𝑦 2 − 𝑦 + 7 = 0 4𝑥 2 − 3𝑥 = 𝑦 − 4𝑦 2 + 25 3𝑦 2 −𝑥 2 = 𝑥 + 7𝑦 − 4 2𝑥 𝑥 + 4 + 36 = 𝑦 − 7 4𝑥 2 − 5x + 23 = 2(10 − 2𝑦 2 ) CIRCLE  What is Circle? A circle is consist of all points on the plane equidistant from a RADIUS fixed point called the center. CENTER The distance from the center to any point on the circle is constant and is called the radius of the circle. 25  STANDARD FORM (𝒙 − 𝒉) 𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓 𝟐 𝟐 𝟐 𝟐 𝒙 +𝒚 =𝒓 ℎ & 𝑘 = 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 r = radius x & 𝑦 = 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑜𝑓 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 26  GENERAL EQUATION 𝑨𝒙 𝟐 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 𝐴𝑥 2& = quadratic terms 𝐶𝑦 2 𝐷𝑥 + 𝐸𝑦 = linear term F =any constant 27 Find the center and the radius of the equation. 𝒙 𝟐 + 𝒚𝟐 = 𝟑𝟔 center = (0,0) radius = 6 28 2023 CIRCLE 29 Find the center and the radius of the equation. (𝒙 + 𝟏)𝟐 + (𝒚 − 𝟐)𝟐 =𝟗 center = (-1 , 2) radius = 3 30 Find the center and the radius of the equation. (𝒙 − 𝟒)𝟐 + (𝒚 − 𝟕)𝟐 = 𝟐𝟓 center = (4 , 7) radius = 5 31  𝟐 𝟐 𝒙 + 𝒚 = 𝟑𝟔 𝟐 𝟐 (𝒙 + 𝟕) +(𝒚 + 𝟖) = 𝟔𝟒 Find the center 𝟐 𝟐 (𝒙 + 𝟏𝟎) +(𝒚 + 𝟗) = 𝟐𝟓 and radius then graph. 𝟐 𝟐 (𝒙 + 𝟓) +(𝒚 − 𝟏𝟎) = 𝟗 (𝒙 + 𝟏)𝟐 +(𝒚 − 𝟐)𝟐 = 𝟏𝟔  𝟐 𝟐 (𝒙 − 𝟑) + 𝒚 = 𝟏𝟔 𝐺𝑒𝑡 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟 𝑥. 𝐺𝑒𝑡 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟 𝑦. 2 (𝑥 − 3) = 0 𝑦2 = 0 𝑟 2 = 16 (𝑥 − 3)2 = 0 𝑦2 = 0 𝑟 2 = 16 𝑥−3=0 𝑦=0 𝑟=4 𝑥=3 𝑪 = (𝟑, 𝟎) 𝒓=𝟒 33 CIRCLE 2023  (𝒙 + 𝟐 𝟏𝟏) +(𝒚 − 𝟐 𝟖) = 𝟔𝟒 𝑪𝒆𝒏𝒕𝒆𝒓 = (−𝟏𝟏, 𝟖) 𝒓𝒂𝒅𝒊𝒖𝒔 = 𝟖 34 CIRCLE 2023  (𝒙 − 𝟐 𝟓) −𝟗 = −(𝒚 + 𝟒)𝟐 (𝒙 𝟐 − 𝟓) +(𝒚 + 𝟐 𝟒) = 𝟗 𝑪𝒆𝒏𝒕𝒆𝒓 = (𝟓, −𝟒) 𝒓𝒂𝒅𝒊𝒖𝒔 = 𝟑 35 CIRCLE 2023  (𝒙 − 𝟐 𝟐) −𝟖 = −(𝒚 − 𝟓)𝟐 (𝒙 𝟐 − 𝟐) +(𝒚 − 𝟐 𝟓) = 𝟖 𝑪𝒆𝒏𝒕𝒆𝒓 = (𝟐, 𝟓) 𝒓𝒂𝒅𝒊𝒖𝒔 = 𝟖 36 CIRCLE 2023  REDUCING CIRCLE’S EQUATION INTO STANDARD FORM 37  STANDARD FORM (𝒙 − 𝒉) 𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓 𝟐 𝟐 𝟐 𝟐 𝒙 +𝒚 =𝒓 ℎ & 𝑘 = 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 r = radius x & 𝑦 = 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑜𝑓 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 38  Find the equation having the Center at (0,0). Center at the origin and radius of 2 units. center = (0,0) 𝟐 𝟐 𝟐 𝒙 + 𝒚 = 𝒓 r=2 𝟐 𝟐 𝟐 𝒙 +𝒚 =𝟐 𝒙 𝟐 + 𝒚𝟐 =𝟒 39 CIRCLE 2023  Find the equation center at the origin and radius of 7 units. center = (0,0) 𝟐 𝟐 𝟐 𝒙 + 𝒚 = 𝒓 r=7 𝟐 𝟐 𝟐 𝒙 +𝒚 =𝟕 𝒙 𝟐 + 𝒚𝟐 = 𝟒𝟗 40 CIRCLE 2023  STANDARD FORM (𝒙 − 𝒉) 𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓 𝟐 𝟐 𝟐 𝟐 𝒙 +𝒚 =𝒓 ℎ & 𝑘 = 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 r = radius x & 𝑦 = 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑜𝑓 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 41  Write the equation of a circle centered at the (2, 4) and radius of 5 units. 𝟐 𝟐 𝟐 h=2 (𝒙 − 𝒉) + (𝒚 − 𝒌) = 𝒓 k=4 𝟐 𝟐 𝟐 r=5 (𝒙 − 𝟐 ) + (𝒚 − 𝟒) = 𝟓 𝟐 𝟐 (𝒙 − 𝟐 ) + (𝒚 − 𝟒) = 𝟐𝟓 42 CIRCLE 2023  Find the equation of a circle centered at the (-7, 5) and radius of 10 units. 𝟐 𝟐 𝟐 h = -7 (𝒙 − 𝒉) + (𝒚 − 𝒌) = 𝒓 k=5 𝟐 𝟐 𝟐 r = 10 (𝒙 + 𝟕 ) + (𝒚 − 𝟓) = 𝟏𝟎 𝟐 𝟐 (𝒙 + 𝟕 ) + (𝒚 − 𝟓) = 𝟏𝟎𝟎 43 CIRCLE 2023  Find the equation of a circle centered at the (3, -6) and radius of 𝟏𝟏 units. 𝟐 𝟐 𝟐 h=3 (𝒙 − 𝒉) + (𝒚 − 𝒌) = 𝒓 k = -6 𝟐 𝟐 𝟐 r = 11 (𝒙 − 𝟑) + (𝒚 + 𝟔) = ( 𝟏𝟏) 𝟐 𝟐 (𝒙 − 𝟑) + (𝒚 + 𝟔) = 𝟏𝟏 44 CIRCLE 2023  The circle has a center at the origin and passing the point (-3,4). 𝒅= (𝒙𝟐 − 𝒙𝟏 )𝟐 +(𝒚𝟐 − 𝒚𝟏 )𝟐 DISTANCE FORMULA 𝐰𝐡𝐞𝐫𝐞: 𝒙𝟏 , 𝒚𝟏 𝒂𝒓𝒆 𝒙𝟐 , 𝒚𝟐 are the coordinates of the two points (0,0) (-3,4). 45 CIRCLE 2023  The circle has a center at the origin and passing the point (-3,4). 𝒅= (𝒙𝟐 − 𝒙𝟏 )𝟐 +(𝒚𝟐 − 𝒚𝟏 )𝟐 (0,0) (-3,4). 𝒅= (−𝟑 − 𝟎)𝟐 +(𝟒 − 𝟎)𝟐 𝒅= 𝟐 (−𝟑) +(𝟒) 𝟐 𝒅 = 𝟗 + 𝟏𝟔 𝒅 = 𝟐𝟓 𝒅 = 𝟓 𝒖𝒏𝒊𝒕𝒔 46 CIRCLE 2023 REDUCING TO STANDARD FORM 47  𝒚 𝟐 −𝟏𝟒𝒚 + 𝒙𝟐 −𝟒𝒙 + 𝟑𝟕 = 𝟎 𝟐 (𝒙 −𝟒𝒙 + ___) + (𝒚 𝟐 − 𝟏𝟒𝒚 + ___) = −𝟑𝟕 −𝟒 −𝟏𝟒 = (−𝟐)𝟐 = 𝟒 = (−𝟕)𝟐 = 𝟒𝟗 𝟐 𝟐 𝟐 𝟐 (𝒙 −𝟒𝒙 + 𝟒) + (𝒚 − 𝟏𝟒𝒚 + 𝟒𝟗) = −𝟑𝟕 + 𝟒 + 𝟒𝟗 𝟐 𝟐 (𝒙 − 𝟐) + (𝒚 − 𝟕) = 𝟏𝟔 𝟏𝟔 = 𝟒 𝑪𝒆𝒏𝒕𝒆𝒓 = (𝟐 , 𝟕) 𝒓𝒂𝒅𝒊𝒖𝒔 = 𝟒 48 CIRCLE 2023  𝒙𝟐 + 𝒚 𝟐 − 𝟏𝟒𝒙 + 𝟒𝒚 + 𝟒𝟓 = 𝟎 𝟐 (𝒙 −𝟏𝟒𝒙 + ___) + (𝒚 𝟐 + 𝟒𝒚 + ___) = −𝟒𝟓 −𝟏𝟒 𝟒 = (−𝟕)𝟐 = 𝟒𝟗 = (𝟐)𝟐 = 𝟒 𝟐 𝟐 𝟐 𝟐 (𝒙 −𝟏𝟒𝒙 + 𝟒𝟗) + (𝒚 + 𝟒𝒚 + 𝟒) = −𝟒𝟓 + 𝟒𝟗 + 𝟒 𝟐 𝟐 (𝒙 − 𝟕) + (𝒚 + 𝟐) = 𝟖 𝟖= 𝟒 ∙𝟐=𝟐 𝟐 𝑪𝒆𝒏𝒕𝒆𝒓 = (𝟕 , −𝟐) 𝒓𝒂𝒅𝒊𝒖𝒔 = 𝟐 𝟐 49 CIRCLE 2023  𝒙𝟐 + 𝒚 𝟐 + 𝟖𝒙 − 𝟒𝒚 − 𝟓 = 𝟎 𝟐 (𝒙 + 𝟖𝒙 + ___) + (𝒚 𝟐 − 𝟒𝒚 + ___) = 𝟓 𝟖 −𝟒 = (𝟒)𝟐 = 𝟏𝟔 = (−𝟐)𝟐 = 𝟒 𝟐 𝟐 𝟐 𝟐 (𝒙 + 𝟖𝒙 + 𝟏𝟔) + (𝒚 − 𝟒𝒚 + 𝟒) = 𝟓 + 𝟏𝟔 + 𝟒 𝟐 𝟐 (𝒙 + 𝟒) + (𝒚 − 𝟐) = 𝟐𝟓 𝟐𝟓 = 𝟓 𝑪𝒆𝒏𝒕𝒆𝒓 = (−𝟒 , 𝟐) 𝒓𝒂𝒅𝒊𝒖𝒔 = 𝟓 50 CIRCLE 2023  Write the general equation of a circle with center (4, - 1) and radius of 7 units. 𝟐 𝟐 𝟐 (𝒙 − 𝒉) + (𝒚 − 𝒌) = 𝒓 h=4 𝟐 𝟐 𝟐 k = -1 (𝒙 − 𝟒 ) + (𝒚 + 𝟏) = 𝟕 r=7 𝒙 − 𝟒 𝒙 − 𝟒 + 𝒚 + 𝟏 𝒚 + 𝟏 = 𝟒𝟗 𝟐 𝟐 𝒙 − 𝟖𝒙 + 𝟏𝟔 + 𝒚 + 𝟐𝒚 + 𝟏 = 𝟒𝟗 𝟐 𝟐 𝒙 + 𝒚 − 𝟖𝒙 + 𝟐𝒚 + 𝟏𝟕 − 𝟒𝟗 = 𝟎 𝟐 𝟐 𝒙 +𝒚 − 𝟖𝒙 + 𝟐𝒚 − 𝟑𝟐 = 𝟎 51 CIRCLE 2023  Write the general equation of a circle with center (-7, - 5) and radius of 9 units. (𝒙 − 𝒉) 𝟐 + (𝒚 − 𝟐 𝒌) = 𝒓𝟐 h = -7 𝟐 𝟐 𝟐 k = -5 (𝒙 + 𝟕) + (𝒚 + 𝟓) = 𝟗 r=9 𝒙 + 𝟕 𝒙 + 𝟕 + 𝒚 + 𝟓 𝒚 + 𝟓 = 𝟖𝟏 𝟐 𝟐 𝒙 + 𝟏𝟒𝒙 + 𝟒𝟗 + 𝒚 + 𝟏𝟎𝒚 + 𝟐𝟓 = 𝟖𝟏 𝟐 𝟐 𝒙 + 𝒚 + 𝟏𝟒𝒙 + 𝟏𝟎𝒚 + 𝟕𝟒 − 𝟖𝟏 = 𝟎 𝟐 𝟐 𝒙 + 𝒚 + 𝟏𝟒𝒙 + 𝟏𝟎𝒚 − 𝟕 = 𝟎 52 CIRCLE 2023  Find the general equation of a circle with center (6, 11) and radius of 10 units. (𝒙 − 𝒉) 𝟐 + (𝒚 − 𝟐 𝒌) = 𝒓𝟐 h=6 𝟐 𝟐 𝟐 k = 11 (𝒙 − 𝟔) + (𝒚 − 𝟏𝟏) = 𝟏𝟎 r = 10 𝒙 − 𝟔 𝒙 − 𝟔 + 𝒚 − 𝟏𝟏 𝒚 − 𝟏𝟏 = 𝟏𝟎𝟎 𝟐 𝟐 𝒙 − 𝟏𝟐𝒙 + 𝟑𝟔 + 𝒚 − 𝟐𝟐𝒚 + 𝟏𝟐𝟏 = 𝟏𝟎𝟎 𝟐 𝟐 𝒙 + 𝒚 − 𝟏𝟐𝒙 − 𝟐𝟐𝒚 + 𝟏𝟓𝟕 − 𝟏𝟎𝟎 = 𝟎 𝟐 𝟐 𝒙 + 𝒚 − 𝟏𝟐𝒙 − 𝟐𝟐𝒚 + 𝟓𝟕 = 𝟎 53 CIRCLE 2023 PARABOLA WHAT IS A PARABOLA? The second commonly known conic section is the parabola. It is defined as a curve that has points which are equidistance from the fixed point and the given line. The shape of the curve of the parabola is similar to an elongated semi-circle 2023 PARABOLA 55  PARTS OF A PARABOLA 2024 PARABOLA 56 VERTEX (V) is the main point of the parabola which lies at the middle portion of the curve. It is also the midpoint of the fixed point of the curve and the given line. DIRECTRIX (DL) is a line which is outside and parallel to the parabola curve. FOCUS (F) it is another important point of the parabola that is located inside of the curve. The focus is also known as the fixed point of the parabola.  FOCAL DISTANCE (a) is the length of space from the focus to the vertex. It is also the distance between the directrix line and the vertex. focal distance LATUS RECTUM (LR) is a chord that is parallel to the directrix line and intersecting the parabola at the point of focus. The end points of the latus rectum are defined as R1 and R2. The length of the latus rectum is always 4 times of the focal distance. The half of the latus rectum is called the semi latus rectum. AXIS OF THE PARABOLA also known as axis of symmetry because it is the line that divides the parabola into two equal parts. The axis of the parabola passes through the points of Vertex and Focus; and this line is perpendicular to the directrix and the latus rectum. axis of symmetry  ECCENTRICITY (e) is the ratio of the distances from a fixed point to one of the points of the curve and from this point to the directrix line. Eccentricity value expresses the degree of roundness of the given curve. For a parabola, eccentricity value is always equal to 1. eccentricity value  STANDARD FORM AND ANALYTICAL PROPERTY OF PARABOLA 2023 PARABOLA 64  IF THE VERTEX IS AT THE ORIGIN (0,0) 𝟐 𝟐 𝒚 = 𝟒𝒑𝒙 𝒙 = 𝟒𝒑𝒚 Horizontal Parabola Vertical Parabola If 4p >0 , opens to the If 4p >0 , opens upward right If 4p 0 , opens upward right If 4p