CHAPTER 12
12-1 INTRODUCTION TO CONIC SECTIONS
OBJECTIVES
RECOGNIZE CONIC SECTIONS AS INTERSECTIONS OF PLANES AND CONES.
USE THE DISTANCE AND MIDPOINT FORMULAS TO SOLVE PROBLEMS.
INTRODUCTION
• IN CHAPTER 5, YOU STUDIED THE PARABOLA. THE PARABOLA IS ONE OF A
FAMILY OF CURVES CALLED CONIC SECTIONS. CONIC SECTIONS ARE
FORMED BY THE INTERSECTION OF A DOUBLE RIGHT CONE AND A PLANE.
THERE ARE FOUR TYPES OF CONIC SECTIONS: CIRCLES, ELLIPSES,
HYPERBOLAS, AND PARABOLAS.
INTRODUCTION
• ALTHOUGH THE PARABOLAS YOU STUDIED IN CHAPTER 5 ARE FUNCTIONS,
MOST CONIC SECTIONS ARE NOT. THIS MEANS THAT YOU OFTEN MUST USE
TWO FUNCTIONS TO GRAPH A CONIC SECTION ON A CALCULATOR.
CIRCLE
• A CIRCLE IS DEFINED BY ITS CENTER AND ITS RADIUS. AN ELLIPSE, AN
ELONGATED SHAPE SIMILAR TO A CIRCLE, HAS TWO PERPENDICULAR AXES
OF DIFFERENT LENGTHS.
EXAMPLE 1A: GRAPHING CIRCLES AND
ELLIPSES ON A CALCULATOR
• GRAPH EACH EQUATION ON A GRAPHING CALCULATOR. IDENTIFY
EACH CONIC SECTION. THEN DESCRIBE THE CENTER AND
INTERCEPTS.
• (X – 1)2 + (Y – 1)2 = 1
SOLUTION
• STEP 1 SOLVE FOR Y SO THAT THE EXPRESSION CAN BE USED IN A
GRAPHING CALCULATOR.
• (Y – 1)2 = 1 – (X – 1)2
SUBTRACT (X – 1)2 FROM BOTH SIDES.
•
•
TAKE SQUARE ROOT OF BOTH SIDES
SOLUTION
• STEP 2 USE TWO EQUATIONS TO SEE THE COMPLETE GRAPH.
Use a square window on your graphing calculator for an
accurate graph. The graphs meet and form a complete circle,
even though it might not appear that way on the calculator.
The graph is a circle with center (1, 1) and intercepts (1,0) and
(0, 1).
EXAMPLE 1B: GRAPHING CIRCLES AND
ELLIPSES ON A CALCULATOR
• 4X2 + 25Y2 = 100
• SOLUTION
• STEP 1 SOLVE FOR Y SO THAT THE EXPRESSION CAN BE USED IN A
GRAPHING CALCULATOR.
• 25Y2 = 100 – 4X2 SUBTRACT 4X2 FROM BOTH SIDES.
2
y =
•
100 – 4x2
25
DIVIDE BOTH SIDES BY 25.
SOLUTION
• STEP 2 USE TWO EQUATIONS TO SEE THE COMPLETE GRAPH.
Use a square window on your graphing calculator for an
accurate graph. The graphs meet and form a complete ellipse,
even though it might not appear that way on the calculator.
The graph is an ellipse with center (0, 0) and intercepts (±5, 0)
and (0, ±2).
PARABOLAS
• A PARABOLA IS A SINGLE CURVE, WHEREAS A HYPERBOLA HAS TWO
CONGRUENT BRANCHES. THE EQUATION OF A PARABOLA USUALLY
CONTAINS EITHER AN X2 TERM OR A Y2 TERM, BUT NOT BOTH. THE
EQUATIONS OF THE OTHER CONICS WILL USUALLY CONTAIN BOTH X2 AND
Y2 TERMS.
EXAMPLE 2A: GRAPHING PARABOLAS
AND HYPERBOLAS ON A CALCULATOR
• GRAPH EACH EQUATION ON A GRAPHING CALCULATOR. IDENTIFY
EACH CONIC SECTION. THEN DESCRIBE THE VERTICES AND THE
DIRECTION THAT THE GRAPH OPENS.
y = –1/2 x2
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
y=–
1
2
2
x
SOLUTION
• STEP 2 USE THE EQUATION TO SEE THE COMPLETE GRAPH.
• THE GRAPH IS A PARABOLA WITH VERTEX (0, 0) THAT OPENS DOWNWARD.
EXAMPLE 2B
• GRAPH EACH EQUATION ON A GRAPHING CALCULATOR. IDENTIFY
EACH CONIC SECTION. THEN DESCRIBE THE VERTICES AND THE
DIRECTION THAT THE GRAPH OPENS.
• Y2 – X 2 = 9
SOLUTION
• STEP 1 SOLVE FOR Y SO THAT THE EXPRESSION CAN BE USED IN A
GRAPHING CALCULATOR.
• Y2 = 9 + X2 ADD X2 TO BOTH SIDES.
Step 2 Use two equations to see the complete
graph.
The graph is a hyperbola that opens vertically with
vertices at (0, ±3).
INTRODUCTION
• EVERY CONIC SECTION CAN BE DEFINED IN TERMS OF DISTANCES. YOU
CAN USE THE MIDPOINT AND DISTANCE FORMULAS TO FIND THE CENTER
AND RADIUS OF A CIRCLE.
INTRODUCTION
• BECAUSE A DIAMETER MUST PASS THROUGH THE CENTER OF A CIRCLE,
THE MIDPOINT OF A DIAMETER IS THE CENTER OF THE CIRCLE. THE
RADIUS OF A CIRCLE IS THE DISTANCE FROM THE CENTER TO ANY POINT
ON THE CIRCLE AND EQUAL TO HALF THE DIAMETER.
EXAMPLE 3: FINDING THE CENTER AND
RADIUS OF A CIRCLE
• FIND THE CENTER AND RADIUS OF A CIRCLE THAT HAS A DIAMETER
WITH ENDPOINTS (5, 4) AND (0, –8).
• STEP 1 FIND THE CENTER OF THE CIRCLE.
• USE THE MIDPOINT FORMULA WITH THE ENDPOINTS (5, 4) AND (0, –8).
(
5+ 0 4–8
, 2 )
2
= (2.5, –2)
SOLUTION
• STEP 2 FIND THE RADIUS.
• USE THE DISTANCE FORMULA WITH (2.5, –2) AND (0, –8)
• THE RADIUS OF THE CIRCLE IS 6.5
EXAMPLE
• FIND THE CENTER AND RADIUS OF A CIRCLE THAT HAS A DIAMETER
WITH ENDPOINTS (2, 6) AND (14, 22).
STUDENT GUIDED PRACTICE
• DO EVEN PROBLEMS FROM 2-12 IN YOUR BOOK PAGE 820
HOMEWORK
• DO PROBLEMS 14,16,18,24,26 AND 32 INY OUR BOOK PAGE 820
CLOSURE
• TODAY WE LEARNED ABOUT CONIC SECTIONS
• NEXT CLASS WE ARE GOING TO LEARN MORE ABOUT CIRCLES