Analytic Geometry Introduction to Conics - Lines and Circles
What You Should Learn • Find the inclination of a line. • Find the angle between two lines. • Find the distance between a point and a line. • Recognize a conic as the intersection of a plane and a double-napped cone. • Classify conics from their general equations. • Find the equation of circle given key pieces of information • Find key information about a circle given an equation.
Plan for the day… • Lines • Find the inclination of a line • Find the angle between two lines • Find the distance between a point and a line • Conic Sections • What are conic sections? • Recognizing different conic sections • Circles • Homework
Inclination of a Line We have learned that the graph of the linear equationy = mx + b is anon-vertical line with slope m and y-intercept (0, b). In this format, the slope of a line was described as the rate of change in y with respect to x. In this section, you will look at the slope of a line in terms of the angle of inclination of the line.
Inclination of a Line Every non-horizontal line must intersect the x-axis. The angle formed by such an intersection determines the inclination of the line, as specified in the following definition. Acute Angle Vertical Line Obtuse Angle Horizontal Line Figure 10.1
Inclination of a Line The inclination of a line is related to its slope in the following manner.
Find the inclination of the line 2x – 3y = 6 • Find the slope -3y = -2x + 6 y = 2/3 x – 2 m = 2/3 • tan θ = 2/3 tan-1 2/3 = θ ≈ 33.7o
Find the inclination of the line 2x + 3y = 6 • Find the slope 3y = -2x + 6 y = -2/3 x + 2 m = -2/3 • tan θ = -2/3 tan-1 -2/3 = θ + 180o ≈ -33.7 + 180o ≈ 146.3o
Class Work Page 704 section 10.1 (degrees only) # 9, 13, 16, 20
The Angle Between Two Lines Two distinct lines in a plane are either parallel or intersecting. If they intersect and are non-perpendicular, their intersection forms two pairs of vertical (opposite) angles. One pair is acute and the other pair is obtuse. The smaller of these angles is called the angle between the two lines.
Find the angle between the two lines • Line 1: 2x – y – 4 = 0 • Line 2: 3x + 4y – 12 = 0 • Slope of line 1 = 2 • Slope of line 2 = -3/4
Class Work Page 704 section 10.1 (degrees only) # 23, 29
The Distance Between a Point and a Line Finding the distance between a line and a point not on the line is an application of perpendicular lines. This distance is defined as the length of the perpendicular line segment joining the point and the line.
The Distance Between a Point and a Line Remember that the values of A, B, and C in this distance formula only correspond to the general equation of a line, Ax + By + C =0.
Find the distance between the point and line • Point (4,1) • Line y = 2x + 1 • Put the line in general form: -2x + y – 1 = 0
Class Work Page 704 section 10.1 # 39, 40
The Cones • Picture 2 cones attached at their points. This figure is called a double cone.
Conic Sections • A conic section is formed by the intersection of a plane with the double cone.
Conic Sections • There are four possible conic sections:
General Equation • The general form for the equation of any conic section is: • The conics in this course will not have the Bxy term, so:
Recognizing a Conic AC = 0 A or C is zero - no x2or no y2 term Parabola A = C A is equal to C, same value Circle AC > 0 A and C have the same sign but have different values Ellipse AC < 0 A and C have different signs Hyperbola
Class Work • Page 733 section 10.4 # 41-48 all
Circles Given an standard form of a conic section, If A = C, then the equation is a circle. Standard form of a circle with the center at (0, 0) is: x2 + y2 = r2 where r is the radius. Standard form of a circle with the center at (h, k) and radius of r is:
Changing format for a circle to identify key information General form of a circle (A=C): Standard form of a circle: How can we get from one to another?
Try this: 9. Find the equation of a circle with center at (1, 2) and a point on the circle is (4, 3)
Homework 39 Classwork • Page 704 section 10.1 (degrees only) #9, 13, 16, 20, 23, 29, 39, 40 • Page 733 section 10.4 # 41-48 all Homework • Page 10 # 57-69 odd (don’t sketch)