CONIC SECTIONS Quadratic Relations Parabola Circle Ellipse Hyperbola

Discriminant Using 𝐴𝑥 2 + 𝐵𝑥𝑦+𝐶𝑦 2 +𝐷𝑥+𝐸𝑦+𝐹=0 Circle  𝐵 2 −4𝐴𝐶<0, 𝐵=0, 𝑎𝑛𝑑 𝐴=𝐶 Ellipse  𝐵 2 −4𝐴𝐶<0 𝑎𝑛𝑑 𝐵≠0 𝑜𝑟 𝐴≠𝐶 Parabola  𝐵 2 −4𝐴𝐶=0 Hyperbola  𝐵 2 −4𝐴𝐶>0

What Conic is this? 9𝑥 2 + 4𝑦 2 +36𝑥−24𝑦+36=0 𝑥 2 − 4𝑦 2 +3𝑥−26𝑦−30=0 4𝑥 2 − 9𝑦 2 +18𝑦+3𝑥=0 𝑥 2 + 𝑦 2 −10𝑥−2𝑦=0

Before we start Conics, you need to know how to Complete the Square

What is completing the square used for? Completing the square is used for all those non-factorable problems!! It is used to solve equations for the variable. Used to set up conics in standard form!

Examples of Perfect Square Trinomials

Rule for Completing the Square Notice that Leading Coefficient must be a one. The middle term (the coefficient with the variable x) is divided by two, then squared. This is now a PST! So, it factors into this!

Example: Find the value of c that makes this a PST, then write the expression as the square of a binomial. x2-3x+c b=-3

Example: Set up by completing the square. x2 + 6x – 8 = 0 Standard form? Move the constant over Don’t forget: Whatever you add to one side of an equation, you MUST add to the other side! Write as PTS!!

4(x + 3)2 = 37 5x2 - 10x + 30 = 0 When the L.C. >1, 4x2 + 24x -1=0

Any questions on Completing the Square??

What do you remember from Math 2?? Let’s start CIRCLES!! What do you remember from Math 2?? (x, y) r y x What we found is the equation of a circle from the distance of the origin (center of the circle) to a point on the circle.

**Center: (h, k) Radius: r ** Standard Form Circle with center at the origin (0,0) Standard form of a circle that is translated **Center: (h, k) Radius: r **

Find the radius and graph. Circles Center at the origin Find the radius and graph. x2 + y2 = 36 x2 + y2 = 12 6x2 + 6y2 = 60

Center that is translated Circles Center that is translated Find the center, radius and graph. (x-2)2 + y2 = 16 Center: ________ r: ______ (x+1)2 + (y-3)2 = 4 Center: ________ r: ______ 2(x+3)2 + 2(y+2)2 = 50 Center: ________ r: ______

Graphing a circle in Standard Form!! To write the standard equation of a translated circle, you may need to complete the square. Example: Standard Form!!  Center: (4, 0) r: 3

Another one you ask!?! Ok, here it is!! Write the standard equation for the circle. State the coordinates of its center and give its radius. Then sketch the graph.

Last One!!! Write the standard equation for the circle. State the center and radius.