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PRE-CALCULUS

Conic Sections

Overview

• To know about the basics of Pre-Calculus
• To be familiar about the Conic Sections
• To solve some real life applications of the Conic Sections in the aviation industry

Basics of Pre-Calculus

• Midpoint Formula
• Distance Formula
• Equation Forms
• Quadratic Formula

Midpoint Formula

By words; If a line segment has two- dimensional endpoints at ( 𝑥 1 , 𝑦 1 ) and ( 𝑥 2 , 𝑦 2 ), then the midpoint of the segment has coordinates of ( 𝑥 1 +𝑥 2 2

𝑦 1 +𝑦 2 2

By symbols; Midpoint = 𝑥 1 +𝑥 2 2

𝑦 1 +𝑦 2 2

Midpoint Examples

Find the midpoint of the line segment who has (- 5 , 6 ) and ( 1 , 7 ). Let say that (- 5 , 6 ) is the coordinates for (𝑥 1 , 𝑦 1 ) and ( 1 , 7 ) is the coordinates for (𝑥 2 , 𝑦 2 ). Solution, M = 𝑥 1 +𝑥 2 2 , 𝑦 1 +𝑦 2 2 M = − 5 + 1 2 , 6 + 7 2 midpoint = (− 2 , 13 2 )

Midpoint Examples

Find the coordinates of the midpoint of the segment with endpoints of ( 6 , - 5 , 1 ) and (- 2 , 4 , 0 ). Let say that ( 6 , - 5 , 1 ) is the coordinates for (𝑥 1 , 𝑦 1 , 𝑧 1 ) and (- 2 , 4 , 0 ) is the coordinates for (𝑥 2 , 𝑦 2 , 𝑧 2 ). Solution, M = 𝑥 1 +𝑥 2 2 , 𝑦 1 +𝑦 2 2 , 𝑧 1 +𝑧 2 2 M = 6 +(− 2 ) 2 , − 5 + 4 2 , 1 + 0 2 midpoint = ( 2 , − 1 2 , 1 2 )

Distance Formula

By words; The distance between two points with coordinates (𝑥 1 , 𝑦 1 ) and (𝑥 2 , 𝑦 2 ) is given. By symbols; 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = (𝑥 2 − 𝑥 1 ) 2 +(𝑦 2 + 𝑦 1 ) 2

Distance Formula

From the Pythagorean Theorem; 𝑎 2 = 𝑏 2

• 𝑐 2 𝑑 2 = 𝑎 2
• 𝑧 2 − 𝑧 1 2 𝑑 2 = 𝑥 2 − 𝑥 1 2
• 𝑦 2 − 𝑦 1 2
• 𝑧 2 − 𝑧 1 2 𝑑 = (𝑥 2 − 𝑥 1 ) 2 +(𝑦 2 − 𝑦 1 ) 2
• 𝑧 2 − 𝑧 1 2

Equation Forms

1. Standard Form is a way of writing an equation which are the variables arranged from greatest to least (depending on the highest degree of the variable and number of variables). It is always equated to a constant and only integers. Ax + By = C Where, A,B,C = constants C= contacts and only integers x, y = variables

Equation Forms

2. General Form is a way of writing an equation similar to the standard form but always equated to 0. Ax + By + C = 0 Where, A,B,C = constants x, y = variables

Quadratic Formula

Given a general quadratic equation of the form 𝑎𝑥 2

• 𝑏𝑥 + 𝑐 = 0 With x representing an unknown, a, b, and c representing constants with a≠0, the quadratic formula is: 𝑥 =

2 − 4𝑎𝑐 2𝑎

Conic Section Families

By definition, the conic sections are the classes of non- degenerate curves generated by the intersections of a plane with one or two nappes of a cone. A conic section can also be realized as the zero set of a quadratic equation in two variables. General Equation of Conic Sections 𝐴𝑥 2

• 𝐵𝑥𝑦 + 𝐶𝑦 2
• 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 Where, A,B,C,D,E, and F are constants

Systems of Non-Linear Equation

1. A linear & a quadratic equation
2. Two equations of the form 𝑎𝑥 2 + 𝑏𝑦 2 = 𝑐
3. Two equations of the form 𝑎𝑥 2 + 𝑏𝑥𝑦 + 𝑐𝑦 2 = 𝑑 Case 1 : if d is not equal to zero, eliminate the constant term of the right side Case 2 : if d= 0 , factor the quadratic equation whose right side is 0
4. Miscellaneous systems a. Reducing the system into equivalent system or simplified equation b. Symmetric equations

A linear & a quadratic equation

Solve for the system: 𝑦^2 + 2 𝑥 = 7 𝑥 − 2 𝑦 = − 7 Through substitution; 𝑥 − 2 𝑦 = − 7 𝑥 = 2 𝑦 − 7 ( 1 ) Substitute equation 1 to 𝑦^2 + 2 𝑥 = 7 𝑦^2 + 2 ( 2 𝑦 − 7 ) = 7 𝑦^2 + 4 𝑦 − 14 − 7 = 0 𝑦^2 + 4 𝑦 − 21 = 0 𝑦 = − 7 & 𝑦 = 3 Substitute the values of y to the equation ( 1 ) If 𝑦 = − 7 𝑥 = 2 𝑦 − 7 𝑥 = 2 (− 7 ) − 7 𝑥 = − 21 If 𝑦 = 3 𝑥 = 2 𝑦 − 7 𝑥 = 2 ( 3 ) − 7 𝑥 = − 1 Then the solutions are (- 21 , 7 ) & (- 1 , 3 )