MATH 5: Review for Chpter 7 Cn ou stte the generl form equtions for the circle, prbol, ellipse, nd hperbol? () Stte the stndrd form eqution for the circle. () Stte the stndrd form eqution for the prbol with verticl is of smmetr. (3) Stte the stndrd form eqution for the ellipse with verticl mjor is. (4) Stte the stndrd form eqution for the hperbol with horizontl trnsverse is. Cn ou grph circle b hnd given the eqution of the circle? (5) Find the center nd rdius of the circle 6 0. Grph the circle b hnd. Cn ou find the eqution of circle given informtion bout the circle? (6) Find the stndrd form eqution of the circle with endpoints of dimeter t,4 nd 3,. Cn ou grph prbol b hnd given the eqution of the prbol? (7) Find the verte, focus, is of smmetr, nd directri of the prbol 6 4 0. Grph the prbol b hnd. Cn ou find the eqution of prbol given informtion bout the prbol? (8) Find the stndrd form eqution of the prbol with verte,5 nd focus 4,5. Cn ou grph n ellipse b hnd given the eqution of the ellipse? (9) Find the center, foci, vertices, length of the mjor is, nd length of the minor is of the ellipse 3 9 0. Grph the ellipse b hnd. Cn ou find the eqution of n ellipse given informtion bout the ellipse? (0) Find the stndrd form eqution of the ellipse with center 0,0, focus 0,, nd verte 0,5. Cn ou grph hperbol b hnd given the eqution of the hperbol? () Find the center, vertices, foci, nd smptotes of the hperbol 4 4 0. Grph the hperbol nd smptotes b hnd. Revised Jnur 06
Cn ou find the eqution of hperbol given informtion bout the hperbol? () Find the stndrd form eqution of the hperbol with vertices,0 4,0. nd,0 nd focus t Cn ou do ppliction problems involving the conic sections? (3) A stellite dish is shped like prboloid. The signls from stellite strike the dish nd re ll reflected to the focl point. The dish is 4 feet cross nd.5 feet deep t its center. Where should the signl receiver be plced? (4) An rch in the shpe of hlf of n ellipse is used to support bridge tht spns river 30 meters wide. The center of the rch is 8 meters bove the river. How high is the rch 0 meters horizontll from the center? Cn ou grph the conic sections on our grphing clcultor? (5) Use our grphing clcultor to grph the prbol 6 5 0 nd find the verte. (6) Use our grphing clcultor to grph the hperbol nd its smptotes. 6 4 Cn ou write the stndrd form of the eqution for the grph of ech conic section? (7) (8) Revised Jnur 06
Answer: ( - ) 36 ( + ) + = 9 foci t ( + 3 3, - ) nd ( - 3 3, - ) Eplntion: (9) Find the stndrd (0) form of the eqution of the hperbol. 4) 0 9 8 7 6 5 4 3-0-9-8 -7-6 -5-4 -3 - - 3 4 5 6 7 8 9 0 - -3-4 -5-6 -7-8 -9-0 Answer: () () Eplntion: 4-5 = (3) Revised Jnur 06 3
Cn ou grphicll solve sstem of nonliner equtions on our grphing clcultor? 3 (4) Use our grphing clcultor to solve: 0 (5) Use our grphing clcultor to solve: 6 8 Cn ou use the lgebric methods of substitution or elimintion b ddition to solve sstem of nonliner equtions? 3 (6) Use substitution to solve: 0 (7) Use elimintion b ddition to solve: 6 8 Revised Jnur 06 4
Answers (with some eplntions): h k r. center: h, k rdius: r. h 4p k verte: h, k focus: h, k p is of smmetr: directri: k p h 3. ( h) ( k) b center: h, k length of mjor is: length of minor is: b distnce between the foci: c, where c b 4. ( h) ( k) b center: h, k distnce between the foci: c, where c b equtions of the smptotes: k b ( h) 5. The stndrd form is: ( 3) ( ) 0 h, k 3, center: rdius: r 0 3.6 Revised Jnur 06 5
3 4 6. h, k,, r (3 ) ( ) 3 The eqution in stndrd form is ( ) ( ) 3 7. 6 4 0 3 4 p, prbol opens up verte: 3, focus: 3, is of smmetr: 3 directri: 3 8. k 4p h h, k,5 p The eqution in stndrd form is 5 8 9. 3 9 0 0 3 h, k 0, center: 3 3.7 b b c b c.4 mjor is: 3 3.4 minor is: b foci:, ;, vertices: 3, ; 3, Revised Jnur 06 6
0. h, k 0,0 5; c c b 4 5 b b h k b 0 0 5 This is the eqution of the ellipse in stndrd form.. 4 4 0 center: h, k, vertices:,,3 nd foci:,,3.4 nd,,0.6 smptotes:. h, k 0,0 c 4 4 c b 6 4 b b h k 0 0 b 4 This is the eqution in stndrd form. Revised Jnur 06 7
3. Hint: Set up the eqution of prbol nd find the focus, becuse the focus is where the receiver is plced. h, k 0,0 h 4p k 0 4p 0 4p,,.5 4p.5 p 3 The receiver should be plced /3 ft. bove the center of the dish. 4. Hint: Set up the eqution of the ellipse nd find when 0. h, k 0,0 5 b 8 h k b 0 0 5 8 5 64 00 5 64 Solve for : 5.96 meters. 5. Solve the eqution for. 3 4 3 4 Ke in the two equtions: 3 4 B trcing nd zooming in, the verte is pproimtel 4,3. Revised Jnur 06 8
6. Solve the eqution for. 6 Ke in the two equtions: 6 6 Find the equtions of the two smptotes. k b h 0 0 4 Ke in the two equtions for the smptotes: 4 You should see the grph of the hperbol bounded b its two smptotes. 3 (7) Center: (,3), Rdius: 5 (8) Center: (0,0), =6, b=5 (9) Center: (,-), =6, b=3 (0) Center: (0,0), =, b=5 Revised Jnur 06 9
() Center: (-,-), =, b=3 () Verte: (-3,-3),Focus:(-,-3), Directri: =-4, P= (3) Verte: (6,5), Focus: (6,8), Directri: =, p= (4) Before ou use our grphing clcultor, ppl our knowledge of conic sections to understnd the problem. 3 3 This is line with positive slope 3 nd -intercept 0,. 0 0 0 This is prbol with verte t the origin, nd it opens up. Hence, ou should see n incresing line intersect prbol in t most two points. The clcultor solution is: ) Solve ech eqution for. 3 ) Ke in ech eqution nd set n pproprite window. 3) Run the intersect progrm to find the two points where the line intersects the prbol. The solution is:,8 nd 0.5,0.5. (5) Before ou use our grphing clcultor, ppl our knowledge of conic sections to understnd the problem. 6 0 0 6 This is circle with center t the origin nd rdius 4. 8 0 8 This is prbol with verte 0, 8, nd it opens up. Hence, ou should see prbol tht opens up intersect circle in t most four points. The clcultor solution is: ) Solve ech eqution for. 6 6 3 8 ) Ke in ech eqution nd set n pproprite window. 3) Run the intersect progrm to find the four points where the prbol intersects the circle. Revised Jnur 06 0
The solution is: 3.,.37 ; 3.,.37.5, 3.37 ;.5, 3.37 (6) Solve 3 for. 3 Substitute this quntit for in the other eqution. 0 3 0 0 0 or 0 or or 8 The solution is, ;,8 Compre this lgebric solution with the grphicl solution in problem (48). (7) 6 6 8 8 8 Solve the resulting qudrtic eqution b the qudrtic formul. 8 0 4 8 33.37 or 3.37 Substitute ech vlue of in the eqution: 6.37 6 3. 3.37 6.5 The solution is 3.,.37 ; 3.,.37 ;.5, 3.37 ;.5, 3.37. Compre this lgebric solution with the grphicl solution in problem (49). Revised Jnur 06