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Calculus
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Chapter 2 / Exercise 11
Calculus
Stewart
Expert Verified
Conic Sections785In Chapter 11, we present several new types of graphs, called conic sections. These include circles, parabolas, ellipses, and hyperbolas. Theseshapes are found in a variety of applications. For example, a reflecting telescope has a mirror whose cross section is in the shape of a parabola,and planetary orbits are modeled by ellipses.As you work through the chapter, you will encounter a variety of equationsassociated with the conic sections. Match the equation with its description,and then use the letter next to each answer to complete the puzzle.111111.1Distance Formula and Circles11.2More on the Parabola11.3The Ellipse and Hyperbola11.4Nonlinear Systems of Equations in Two Variables11.5Nonlinear Inequalities and Systems of Inequalities1.a. standard form of a parabola withhorizontal axis of symmetry 2.o. standard form of an ellipse cen-tered at the origin3.f.standard form of a circle centeredat (h, k)4.t.standard form of a hyperbola withhorizontal transverse axis5.s. standard form of a circle centeredat the origin6.c. standard form of a parabola withvertical axis of symmetry7.r.distance formulax2a2y2b211xh221yk22r2ya1xh22kd21x2x1221y2y122xa1yk22hx2y2r2x2a2y2b21He became a math teacher due to some prime6357142
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Calculus
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Chapter 2 / Exercise 11
Calculus
Stewart
Expert Verified
786Chapter 11Conic Sections1. Distance FormulaSuppose we are given two points and in a rectangular coordi-nate system. The distance between the two points can be found by using thePythagorean theorem (Figure 11-1).First draw a right triangle with the dis-tance das the hypotenuse. The length of thehorizontal leg ais , and the length ofthe vertical leg bis From thePythagorean theorem we havePythagorean theoremBecause distance is positive, reject thenegative value.The Distance FormulaThe distance dbetween the points (x1,y1) and (x2,y2) is d21x2x1221y2y12221x2x1221y2y122d21x2x1221y2y1221x2x1221y2y122d2a2b20y2y10.0x2x101x2, y221x1, y12Figure 11-1Finding the Distance Between Two PointsFind the distance between the points and (Figure 11-2).Solution:andandLabel the points.Apply the distance formula.1.Find the distance between the points (4,2) and (2,5).Skill Practice2113222131521361621622142223 142122 423 112132 42d21x2x1221y2y1221x2, y221x1, y1214, 1212, 3214, 1212, 32Example 1b0y2y10a0x2x10(x1, y1)(x2, y2)dxyFigure 11-2yx(2, 3)(4, 1)d2 13 7.21Section 11.1Distance Formula and CirclesConcepts1.Distance Formula2.Circles3.Writing an Equation of a CircleSkill Practice Answers1.315
Section 11.1Distance Formula and Circles787TIP:The order in which the points are labeled does not affect the result ofthe distance formula. For example, if the points in Example 1 had been labeledin reverse, the distance formula would still yield the same result:andand21131521361621622142223 122142 4233112 421x1, y12

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