MATH 115: Review for Chapter 7

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Frederick Edward Stone
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1 MATH 5: Review for Chpter 7 Cn you stte the generl form equtions for the circle, prbol, ellipse, nd hyperbol? () Stte the stndrd form eqution for the circle. () Stte the stndrd form eqution for the prbol with verticl xis of symmetry. (3) Stte the stndrd form eqution for the ellipse with verticl mjor xis. (4) Stte the stndrd form eqution for the hyperbol with horizontl trnsverse xis. Cn you grph circle by hnd given the eqution of the circle? (5) Find the center nd rdius of the circle x y 6x y = 0. Grph the circle by hnd. Cn you find the eqution of circle given informtion bout the circle? (6) Find the stndrd form eqution of the circle with endpoints of, 4 3,. dimeter t ( ) nd ( ) Cn you grph prbol by hnd given the eqution of the prbol? (7) Find the vertex, focus, xis of symmetry, nd directrix of the prbol x 6x 4y = 0. Grph the prbol by hnd. Cn you find the eqution of prbol given informtion bout the prbol? (8) Find the stndrd form eqution of the prbol with vertex (, 5 ) nd focus ( 4, 5 ). Cn you grph n ellipse by hnd given the eqution of the ellipse? (9) Find the center, foci, vertices, length of the mjor xis, nd length of the minor xis of the ellipse x 3y y 9 = 0. Grph the ellipse by hnd. Cn you 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 vertex ( 0, 5 ). Cn you grph hyperbol by hnd given the eqution of the hyperbol? () Find the center, vertices, foci, nd symptotes of the hyperbol y x 4y 4x = 0. Grph the hyperbol nd symptotes by hnd.

2 Cn you find the eqution of hyperbol given informtion bout the hyperbol? () Find the stndrd form eqution of the hyperbol with vertices, 0 4, 0. ( ) nd (, 0 ) nd focus t ( ) Cn you 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 horizontlly from the center? Cn you grph the conic sections on your grphing clcultor? (5) Use your grphing clcultor to grph the prbol y 6y x 5 = 0 nd find the vertex. (6) Use your grphing clcultor to grph the hyperbol nd its symptotes. x y = 6 4

3 Answers (with some explntions):. ( ) ( ) center: ( h, k ) rdius: r x h y k = r. ( x h) = 4p ( y k) vertex: ( h, k ) focus: ( h, k p) xis of symmetry: x directrix: y = k p = h 3. ( x h) ( y k) b = center: ( h, k ) length of mjor xis: length of minor xis: b distnce between the foci: c, where c = b 4. ( x h) ( y k) b = center: ( h, k ) distnce between the foci: c, where c = b equtions of the symptotes: y k = ± b ( x h) 5. The stndrd form is: ( x 3) ( y ) = 0 center: ( h, k ) = ( 3, ) rdius: r = 0 3.6

4 ( h, k) =, = (,) r = (3 ) ( ) = 3 The eqution in stndrd form is ( x ) ( y ) = 3 7. x x y 6 4 = 0 ( x 3) = 4 ( y ) p =, prbol opens up vertex: ( 3, ) focus: ( 3, ) xis of symmetry: x = 3 directrix: y = 3 8. ( y k) = 4p ( x h) ( h, k ) = (,5) p = The eqution in stndrd form is ( y 5) = 8( x )

5 9. x y y 3 9 = 0 ( x 0) ( y ) = 3 h, k = 0, center: ( ) ( ) = 3 = 3.7 b = b = c = b = c =.4 mjor xis: = minor xis: b = foci: (, ); (, ) vertices: ( 3, ); ( 3, ) 0. ( h, k ) = ( 0,0) = 5; c = c = b 4 = 5 b b = ( x h) ( y k) = b ( x 0) ( y 0) = 5 This is the eqution of the ellipse in stndrd form.

6 . y x y x 4 4 = 0 ( y ) ( x ) = h, k =,, center: ( ) ( ) vertices: (, 3 ) nd ( ) foci: (, ) (,3.4) nd (, ) (,0.6 ) symptotes: y = ± ( x ). ( h, k ) = ( 0,0) 4 = = c = 4 c = b 6 = 4 b b = ( x h) ( y k) = b ( x 0) ( y 0) = 4 This is the eqution in stndrd form.

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) ( x h) = 4p ( y k) ( x 0) = 4p ( y 0) x = 4py ( x, y) = (,.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 y when x = 0.

8 ( h, k) = ( 0,0) = 5 b = 8 ( x h) ( y k) b = ( x 0) ( y 0) 5 8 = x y 5 64 = 00 y 5 64 = Solve for y: y 5.96 meters. 5. Solve the eqution for y. y = 3 ± 4 x Key in the two equtions: y y = 3 4 x = 3 4 x By trcing nd zooming in, the vertex is pproximtely ( ) 4, Solve the eqution for y. x 6 y = ± Key in the two equtions: y y x 6 = x 6 = Find the equtions of the two symptotes. b y k = ± ( x h) y 0 = ± ( x 0) 4 y = ± x

9 y = x 3 Key in the two equtions for the symptotes: y4 = x You should see the grph of the hyperbol bounded by its two symptotes.

MATH 115: Review for Chapter 7

MATH 115: Review for Chapter 7 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