Solving Equations Involving Hyperbolas and Parabolas
Utilize substitution to solve equations involving hyperbolas and parabolas that touch at specific points. Discover the values of variables by manipulating equations and identifying intersections between the curves. Utilize the discriminant to solve for double roots and tangent points effectively.
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Presentation Transcript
The two curves just touch, as shown. What is the value of ? ? ? ? = ??2 ? ?2 ?2= 0.12
? ? ? ? Answers: 1 10 1 8 1 5 1 4 1 2 3 2 1 10 1 8 1 5 1 4 1 2 3 2 5 5 ?2 ?2= ?2 4 4 5 2 5 2 Find your value of ? in the table to locate the appropriate value of ?. 2 2 1 1 1 3 1 6 1 7 1 12 1 14 1 3 1 6 1 7 1 12 1 14 3 3 7 2 7 2 Can t find it? Try this table 6 6 7 7
Solve using substitution. ? = ??2 ?2 ?2= ?2 ?2 ?2?4= ?2 ?2 ?2?2 2= ?2 ?2?2 2 ?2+ ?2= 0 ?2= 1 1 4?2?2 For a double root (tangent) the discriminant = 0, so 4?2?2= 1 ?? =1 2?2 2 1 with intersection at ? 2,?
The two curves just touch, as shown. What is the value of ? ? ? The two curves just touch, as shown. What is the value of ? ? ? ? = ??2 ? = ??2 ? ? ?2 ?2= 0.12 ?2 ?2= 0.22 SIC_20 SIC_20 The two curves just touch, as shown. What is the value of ? ? ? The two curves just touch, as shown. What is the value of ? ? ? ? = ??2 ? = ??2 ? ? ?2 ?2= 0.252 ?2 ?2= 0.52 SIC_20 SIC_20
The two curves just touch, as shown. What is the value of ? ? ? The two curves just touch, as shown. What is the value of ? ? ? ? = ??2 ? = ??2 ? ? ?2 ?2= 12 ?2 ?2= 32 SIC_20 SIC_20 The two curves just touch, as shown. What is the value of ? ? ? The two curves just touch, as shown. What is the value of ? ? ? ? = ??2 ? = ??2 ? ? ?2 ?2= 62 ?2 ?2= 52 SIC_20 SIC_20
The two curves just touch, as shown. What is the value of ? ? ? The two curves just touch, as shown. What is the value of ? ? ? ? = ??2 ? = ??2 ? ? ?2 ?2= 72 ?2 ?2= 3.52 SIC_20 SIC_20 The two curves just touch, as shown. What is the value of ? ? ? The two curves just touch, as shown. What is the value of ? ? ? ? = ??2 ? = ??2 ? ? ?2 ?2= 42 ?2 ?2= 1.52 SIC_20 SIC_20
