1. ਕੋਸ਼ਿਸ਼ ਕਰੋ
2. x 2 + y 2 = 16
3. (x−1) 2 / 16 + (y−3) 2 / 4 = 1
4. ਕੇਂਦਰ: (0,0); ਸਿਖਰ: (±6,0); ਸਹਿ-ਸਿਖਰ: (0,±2); ਨਾਭੀਆਂ: (±4√2,0)
5. ਮਿਆਰੀ ਰੂਪ: x 2 / 16 + y 2 / 49 = 1; ਕੇਂਦਰ: (0,0); ਸਿਖਰ: (0,±7); ਸਹਿ-ਸਿਖਰ: (±4,0); ਨਾਭੀਆਂ: (0,±√33)
6. ਕੇਂਦਰ: (4,2); ਸਿਖਰ: (−2,2) ਅਤੇ (10,2); ਸਹਿ-ਸਿਖਰ: (4,2−2√5) ਅਤੇ (4,2+2√5); ਨਾਭੀਆਂ: (0,2) ਅਤੇ (8,2)
7. (x−3) 2 / 4 + (y+1) 2 / 16 = 1; ਕੇਂਦਰ: (3,−1); ਸਿਖਰ: (3,−5) ਅਤੇ (3,3); ਸਹਿ-ਸਿਖਰ: (1,−1) ਅਤੇ (5,−1); ਨਾਭੀਆਂ: (3,−1−2√3) ਅਤੇ (3,−1+2√3)
8. ⓐ x 2 / 57,600 + y 2 / 25,600 = 1
9. ⓑ ਲੋਕ 358 ਫੁੱਟ ਦੀ ਦੂਰੀ 'ਤੇ ਖੜ੍ਹੇ ਹਨ।
10. ਸਿਖਰ: (±3,0); ਨਾਭੀਆਂ: (±√34,0)
11. y 2 / 4 - x 2 / 16 = 1
12. (y−3) 2 / 25 - (x−1) 2 / 144 = 1
13. ਸਿਖਰ: (±12,0); ਸਹਿ-ਸਿਖਰ: (0,±9); ਨਾਭੀਆਂ: (±15,0); ਅਨੰਤ ਸਪਰਸ਼ ਰੇਖਾਵਾਂ: y=±3/4 x;
14. ਕੇਂਦਰ: (3,−4); ਸਿਖਰ: (3,−14) ਅਤੇ (3,6); ਸਹਿ-ਸਿਖਰ: (−5,−4) ਅਤੇ (11,−4); ਨਾਭੀਆਂ: (3,−4−2√41) ਅਤੇ (3,−4+2√41); ਅਨੰਤ ਸਪਰਸ਼ ਰੇਖਾਵਾਂ: y=±5/4 (x−3)−4
15. ਟਾਵਰ ਦੀਆਂ ਸਾਈਡਾਂ ਨੂੰ ਹਾਈਪਰਬੋਲਿਕ ਸਮੀਕਰਨ ਦੁਆਰਾ ਮਾਡਲ ਕੀਤਾ ਜਾ ਸਕਦਾ ਹੈ। x 2 / 400 - y 2 / 3600 = 1 ਜਾਂ x 2 / 20 2 - y 2 / 60 2 = 1.
16. ਨਾਭੀ: (−4,0); ਨਿਰਦੇਸ਼ਕ ਰੇਖਾ: x=4; ਲੇਟਸ ਰੈਕਟਮ ਦੇ ਸਿਰੇ: (−4,±8)
17. ਨਾਭੀ: (0,2); ਨਿਰਦੇਸ਼ਕ ਰੇਖਾ: y=−2; ਲੇਟਸ ਰੈਕਟਮ ਦੇ ਸਿਰੇ: (±4,2).
18. x 2 = 14y.
19. ਸਿਖਰ: (8,−1); ਸਮਰੂਪਤਾ ਧੁਰਾ: y=−1; ਨਾਭੀ: (9,−1); ਨਿਰਦੇਸ਼ਕ ਰੇਖਾ: x=7; ਲੇਟਸ ਰੈਕਟਮ ਦੇ ਸਿਰੇ: (9,−3) ਅਤੇ (9,1).
20. ਸਿਖਰ: (−2,3); ਸਮਰੂਪਤਾ ਧੁਰਾ: x=−2; ਨਾਭੀ: (−2,−2); ਨਿਰਦੇਸ਼ਕ ਰੇਖਾ: y=8; ਲੇਟਸ ਰੈਕਟਮ ਦੇ ਸਿਰੇ: (−12,−2) ਅਤੇ (8,−2).
21. ⓐ y 2 = 1280x
22. ⓑ ਕੁੱਕਰ ਦੀ ਡੂੰਘਾਈ 500 ਮਿਲੀਮੀਟਰ ਹੈ।
23. ⓐ ਹਾਈਪਰਬੋਲਾ
24. ⓑ ਇੱਲਿਪਸ
x ′ 2 4 + y ′ 2 1 =1 x ′ 2 4 + y ′ 2 1 =1
ⓐ hyperbola
ⓑ ellipse
ellipse; e= 1 3 ;x=−2 e= 1 3 ;x=−2
r= 1 1−cosθ r= 1 1−cosθ
4−8x+3 x 2 − y 2 =0 4−8x+3 x 2 − y 2 =0
12.1 Section Exercises
An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.
This special case would be a circle.
It is symmetric about the x-axis, y-axis, and the origin.
yes; x 2 3 2 + y 2 2 2 =1 x 2 3 2 + y 2 2 2 =1
yes; x 2 ( 1 2 ) 2 + y 2 ( 1 3 ) 2 =1 x 2 ( 1 2 ) 2 + y 2 ( 1 3 ) 2 =1
x 2 2 2 + y 2 7 2 =1; x 2 2 2 + y 2 7 2 =1; Endpoints of major axis ( 0,7 ) ( 0,7 ) and ( 0,−7 ). ( 0,−7 ). Endpoints of minor axis ( 2,0 ) ( 2,0 ) and ( −2,0 ). ( −2,0 ). Foci at ( 0,3 5 ),( 0,−3 5 ). ( 0,3 5 ),( 0,−3 5 ).
x 2 ( 1 ) 2 + y 2 ( 1 3 ) 2 =1; x 2 ( 1 ) 2 + y 2 ( 1 3 ) 2 =1; Endpoints of major axis ( 1,0 ) ( 1,0 ) and ( −1,0 ). ( −1,0 ). Endpoints of minor axis ( 0, 1 3 ),( 0,− 1 3 ). ( 0, 1 3 ),( 0,− 1 3 ). Foci at ( 2 2 3 ,0 ),( − 2 2 3 ,0 ). ( 2 2 3 ,0 ),( − 2 2 3 ,0 ).
( x−2 ) 2 7 2 + ( y−4 ) 2 5 2 =1; ( x−2 ) 2 7 2 + ( y−4 ) 2 5 2 =1; Endpoints of major axis ( 9,4 ),( −5,4 ). ( 9,4 ),( −5,4 ). Endpoints of minor axis ( 2,9 ),( 2,−1 ). ( 2,9 ),( 2,−1 ). Foci at ( 2+2 6 ,4 ),( 2−2 6 ,4 ). ( 2+2 6 ,4 ),( 2−2 6 ,4 ).
( x+5 ) 2 2 2 + ( y−7 ) 2 3 2 =1; ( x+5 ) 2 2 2 + ( y−7 ) 2 3 2 =1; Endpoints of major axis ( −5,10 ),( −5,4 ). ( −5,10 ),( −5,4 ). Endpoints of minor axis ( −3,7 ),( −7,7 ). ( −3,7 ),( −7,7 ). Foci at ( −5,7+ 5 ),( −5,7− 5 ). ( −5,7+ 5 ),( −5,7− 5 ).
( x−1 ) 2 3 2 + ( y−4 ) 2 2 2 =1; ( x−1 ) 2 3 2 + ( y−4 ) 2 2 2 =1; Endpoints of major axis ( 4,4 ),( −2,4 ). ( 4,4 ),( −2,4 ). Endpoints of minor axis ( 1,6 ),( 1,2 ). ( 1,6 ),( 1,2 ). Foci at ( 1+ 5 ,4 ),( 1− 5 ,4 ). ( 1+ 5 ,4 ),( 1− 5 ,4 ).
( x−3 ) 2 ( 3 2 ) 2 + ( y−5 ) 2 ( 2 ) 2 =1; ( x−3 ) 2 ( 3 2 ) 2 + ( y−5 ) 2 ( 2 ) 2 =1; Endpoints of major axis ( 3+3 2 ,5 ),( 3−3 2 ,5 ). ( 3+3 2 ,5 ),( 3−3 2 ,5 ). Endpoints of minor axis ( 3,5+ 2 ),( 3,5− 2 ). ( 3,5+ 2 ),( 3,5− 2 ). Foci at ( 7,5 ),( −1,5 ). ( 7,5 ),( −1,5 ).
( x+5 ) 2 ( 5 ) 2 + ( y−2 ) 2 ( 2 ) 2 =1; ( x+5 ) 2 ( 5 ) 2 + ( y−2 ) 2 ( 2 ) 2 =1; Endpoints of major axis ( 0,2 ),( −10,2 ). ( 0,2 ),( −10,2 ). Endpoints of minor axis ( −5,4 ),( −5,0 ). ( −5,4 ),( −5,0 ). Foci at ( −5+ 21 ,2 ),( −5− 21 ,2 ). ( −5+ 21 ,2 ),( −5− 21 ,2 ).
( x+3 ) 2 ( 5 ) 2 + ( y+4 ) 2 ( 2 ) 2 =1; ( x+3 ) 2 ( 5 ) 2 + ( y+4 ) 2 ( 2 ) 2 =1; Endpoints of major axis ( 2,−4 ),( −8,−4 ). ( 2,−4 ),( −8,−4 ). Endpoints of minor axis ( −3,−2 ),( −3,−6 ). ( −3,−2 ),( −3,−6 ). Foci at ( −3+ 21 ,−4 ),( −3− 21 ,−4 ). ( −3+ 21 ,−4 ),( −3− 21 ,−4 ).
Foci ( −3,−1+ 11 ),( −3,−1− 11 ) ( −3,−1+ 11 ),( −3,−1− 11 )
Focus ( 0,0 ) ( 0,0 )
Foci ( −10,30 ),( −10,−30 ) ( −10,30 ),( −10,−30 )
Center ( 0,0 ), ( 0,0 ), Vertices ( 4,0 ),( −4,0 ),(0,3),(0,−3), ( 4,0 ),( −4,0 ),(0,3),(0,−3), Foci ( 7 ,0 ),( − 7 ,0 ) ( 7 ,0 ),( − 7 ,0 )
ਕੇਂਦਰ (0,0), ਸਿਖਰ (1/9,0), (−1/9,0), (0,1/7), (0,−1/7), ਨਾਭੀਆਂ (0, 4√2/63), (0,−4√2/63)
ਕੇਂਦਰ (−3,3), ਸਿਖਰ (0,3), (−6,3), (−3,0), (−3,6), ਨਾਭੀ (−3,3)
ਨੋਟ ਕਰੋ ਕਿ ਇਹ ਅੰਡਾਕਾਰ ਇੱਕ ਚੱਕਰ ਹੈ। ਚੱਕਰ ਦਾ ਕੇਵਲ ਇੱਕ ਨਾਭੀ ਹੁੰਦੀ ਹੈ, ਜੋ ਕਿ ਕੇਂਦਰ ਨਾਲ ਮੇਲ ਖਾਂਦੀ ਹੈ।
ਕੇਂਦਰ (1,1), ਸਿਖਰ (5,1), (−3,1), (1,3), (1,−1), ਨਾਭੀਆਂ (1+2√3,1), (1−2√3,1)
ਕੇਂਦਰ (−4,5), ਸਿਖਰ (−2,5), (−6,5), (−4,6), (−4,4), ਨਾਭੀਆਂ (−4+√3,5), (−4−√3,5)
ਕੇਂਦਰ (−2,1), ਸਿਖਰ (0,1), (−4,1), (−2,5), (−2,−3), ਨਾਭੀਆਂ (−2,1+2√3), (−2,1−2√3)
ਕੇਂਦਰ (−2,−2), ਸਿਖਰ (0,−2), (−4,−2), (−2,0), (−2,−4), ਨਾਭੀ (−2,−2)
x²/25 + y²/29 = 1
(x−4)²/25 + (y−2)²/1 = 1
(x+3)²/16 + (y−4)²/4 = 1
x²/81 + y²/9 = 1
(x+2)²/4 + (y−2)²/9 = 1
ਖੇਤਰਫਲ = 12π ਵਰਗ ਇਕਾਈਆਂ
ਖੇਤਰਫਲ = 2/5 π ਵਰਗ ਇਕਾਈਆਂ।
ਖੇਤਰਫਲ = 9π ਵਰਗ ਇਕਾਈਆਂ।
x²/(4h²) + y²/(1/4h²) = 1
x²/400 + y²/144 = 1। ਦੂਰੀ = 17.32 ਫੁੱਟ
ਲਗਭਗ 51.96 ਫੁੱਟ
12.2 ਭਾਗ ਅਭਿਆਸ
ਅਤਿਪਰਵਲੈ (hyperbola) ਸਮਤਲ ਵਿੱਚ ਉਹਨਾਂ ਬਿੰਦੂਆਂ ਦਾ ਸਮੂਹ ਹੈ ਜਿਨ੍ਹਾਂ ਦੀਆਂ ਦੋ ਨਿਸ਼ਚਿਤ ਬਿੰਦੂਆਂ (ਨਾਭੀਆਂ) ਤੋਂ ਦੂਰੀਆਂ ਦਾ ਅੰਤਰ ਇੱਕ ਧਨ ਸਥਿਰ ਅੰਕ ਹੁੰਦਾ ਹੈ।
ਨਾਭੀਆਂ ਅਤਿਪਰਵਲੈ ਦੇ ਮੁੱਖ ਅਕਸ਼ ਉੱਤੇ ਸਥਿਤ ਹੋਣੀਆਂ ਚਾਹੀਦੀਆਂ ਹਨ ਅਤੇ ਅਤਿਪਰਵਲੈ ਦੇ ਅੰਦਰੂਨੀ ਭਾਗ ਵਿੱਚ ਹੋਣੀਆਂ ਚਾਹੀਦੀਆਂ ਹਨ।
ਕੇਂਦਰ ਨਾਭੀਆਂ ਨੂੰ ਜੋੜਨ ਵਾਲੇ ਰੇਖਾਖੰਡ ਦਾ ਮੱਧਬਿੰਦੂ ਹੋਣਾ ਚਾਹੀਦਾ ਹੈ।
ਹਾਂ, x²/6² − y²/3² = 1
ਹਾਂ, x²/4² − y²/5² = 1
x 2 5 2 − y 2 6 2 =1; x 2 5 2 − y 2 6 2 =1; vertices: ( 5,0 ),( −5,0 ); ( 5,0 ),( −5,0 ); foci: ( 61 ,0 ),( − 61 ,0 ); ( 61 ,0 ),( − 61 ,0 ); asymptotes: y= 6 5 x,y=− 6 5 x y= 6 5 x,y=− 6 5 x
y 2 2 2 − x 2 9 2 =1; y 2 2 2 − x 2 9 2 =1; vertices: ( 0,2 ),( 0,−2 ); ( 0,2 ),( 0,−2 ); foci: ( 0, 85 ),( 0,− 85 ); ( 0, 85 ),( 0,− 85 ); asymptotes: y= 2 9 x,y=− 2 9 x y= 2 9 x,y=− 2 9 x
( x−1 ) 2 3 2 − ( y−2 ) 2 4 2 =1; ( x−1 ) 2 3 2 − ( y−2 ) 2 4 2 =1; vertices: ( 4,2 ),( −2,2 ); ( 4,2 ),( −2,2 ); foci: ( 6,2 ),( −4,2 ); ( 6,2 ),( −4,2 ); asymptotes: y= 4 3 ( x−1 )+2,y=− 4 3 ( x−1 )+2 y= 4 3 ( x−1 )+2,y=− 4 3 ( x−1 )+2
( x−2 ) 2 7 2 − ( y+7 ) 2 7 2 =1; ( x−2 ) 2 7 2 − ( y+7 ) 2 7 2 =1; vertices: ( 9,−7 ),( −5,−7 ); ( 9,−7 ),( −5,−7 ); foci: ( 2+7 2 ,−7 ),( 2−7 2 ,−7 ); ( 2+7 2 ,−7 ),( 2−7 2 ,−7 ); asymptotes: y=x−9,y=−x−5 y=x−9,y=−x−5
( x+3 ) 2 3 2 − ( y−3 ) 2 3 2 =1; ( x+3 ) 2 3 2 − ( y−3 ) 2 3 2 =1; vertices: ( 0,3 ),( −6,3 ); ( 0,3 ),( −6,3 ); foci: ( −3+3 2 ,1 ),( −3−3 2 ,1 ); ( −3+3 2 ,1 ),( −3−3 2 ,1 ); asymptotes: y=x+6,y=−x y=x+6,y=−x
( y−4 ) 2 2 2 − ( x−3 ) 2 4 2 =1; ( y−4 ) 2 2 2 − ( x−3 ) 2 4 2 =1; vertices: ( 3,6 ),( 3,2 ); ( 3,6 ),( 3,2 ); foci: ( 3,4+2 5 ),( 3,4−2 5 ); ( 3,4+2 5 ),( 3,4−2 5 ); asymptotes: y= 1 2 ( x−3 )+4,y=− 1 2 ( x−3 )+4 y= 1 2 ( x−3 )+4,y=− 1 2 ( x−3 )+4
( y+5 ) 2 7 2 − ( x+1 ) 2 70 2 =1; ( y+5 ) 2 7 2 − ( x+1 ) 2 70 2 =1; vertices: ( −1,2 ),( −1,−12 ); ( −1,2 ),( −1,−12 ); foci: ( −1,−5+7 101 ),( −1,−5−7 101 ); ( −1,−5+7 101 ),( −1,−5−7 101 ); asymptotes: y= 1 10 ( x+1 )−5,y=− 1 10 ( x+1 )−5 y= 1 10 ( x+1 )−5,y=− 1 10 ( x+1 )−5
( x+3 ) 2 5 2 − ( y−4 ) 2 2 2 =1; ( x+3 ) 2 5 2 − ( y−4 ) 2 2 2 =1; vertices: ( 2,4 ),( −8,4 ); ( 2,4 ),( −8,4 ); foci: ( −3+ 29 ,4 ),( −3− 29 ,4 ); ( −3+ 29 ,4 ),( −3− 29 ,4 ); asymptotes: y= 2 5 ( x+3 )+4,y=− 2 5 ( x+3 )+4 y= 2 5 ( x+3 )+4,y=− 2 5 ( x+3 )+4
y= 2 5 ( x−3 )−4,y=− 2 5 ( x−3 )−4 y= 2 5 ( x−3 )−4,y=− 2 5 ( x−3 )−4
y= 3 4 ( x−1 )+1,y=− 3 4 ( x−1 )+1 y= 3 4 ( x−1 )+1,y=− 3 4 ( x−1 )+1
x 2 9 − y 2 16 =1 x 2 9 − y 2 16 =1
( x−6 ) 2 25 − ( y−1 ) 2 11 =1 ( x−6 ) 2 25 − ( y−1 ) 2 11 =1
( x−4 ) 2 25 − ( y−2 ) 2 1 =1 ( x−4 ) 2 25 − ( y−2 ) 2 1 =1
y 2 16 − x 2 25 =1 y 2 16 − x 2 25 =1
y 2 9 − ( x+1 ) 2 9 =1 y 2 9 − ( x+1 ) 2 9 =1
( x+3 ) 2 25 − ( y+3 ) 2 25 =1 ( x+3 ) 2 25 − ( y+3 ) 2 25 =1
y( x )=3 x 2 +1 ,y( x )=−3 x 2 +1 y( x )=3 x 2 +1 ,y( x )=−3 x 2 +1
y( x )=1+2 x 2 +4x+5 ,y( x )=1−2 x 2 +4x+5 y( x )=1+2 x 2 +4x+5 ,y( x )=1−2 x 2 +4x+5
x 2 25 − y 2 25 =1 x 2 25 − y 2 25 =1
x 2 100 − y 2 25 =1 x 2 100 − y 2 25 =1
x 2 400 − y 2 225 =1 x 2 400 − y 2 225 =1
4(x-1)2-y22=16 4(x-1)2-y22=16
( x−h ) 2 a2 - (y-k)2 b2 =(x-3)2-9y2=4 ( x−h ) 2 a2 -(y-k)2 b2=(x-3)2-9y2=4
12.3 Section Exercises
A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.
The graph will open down.
The distance between the focus and directrix will increase.
yes x2=4(116)y x2=4(116)y
yes ( y−3 ) 2 =4(2)( x−2 ) ( y−3 ) 2 =4(2)( x−2 )
y 2 = 1 8 x,V:(0,0);F:( 1 32 ,0 );d:x=− 1 32 y 2 = 1 8 x,V:(0,0);F:( 1 32 ,0 );d:x=− 1 32
x 2 =− 1 4 y,V:( 0,0 );F:( 0,− 1 16 );d:y= 1 16 x 2 =− 1 4 y,V:( 0,0 );F:( 0,− 1 16 );d:y= 1 16
y 2 = 1 36 x,V:( 0,0 );F:( 1 144 ,0 );d:x=− 1 144 y 2 = 1 36 x,V:( 0,0 );F:( 1 144 ,0 );d:x=− 1 144
( x−1 ) 2 =4( y−1 ),V:( 1,1 );F:( 1,2 );d:y=0 ( x−1 ) 2 =4( y−1 ),V:( 1,1 );F:( 1,2 );d:y=0
( y−4 ) 2 =2( x+3 ),V:( −3,4 );F:( − 5 2 ,4 );d:x=− 7 2 ( y−4 ) 2 =2( x+3 ),V:( −3,4 );F:( − 5 2 ,4 );d:x=− 7 2
( x+4 ) 2 =24( y+1 ),V:( −4,−1 );F:( −4,5 );d:y=−7 ( x+4 ) 2 =24( y+1 ),V:( −4,−1 );F:( −4,5 );d:y=−7
( y−3 ) 2 =−12( x+1 ),V:( −1,3 );F:( −4,3 );d:x=2 ( y−3 ) 2 =−12( x+1 ),V:( −1,3 );F:( −4,3 );d:x=2
( x−5 ) 2 = 4 5 ( y+3 ),V:( 5,−3 );F:( 5,− 14 5 );d:y=− 16 5 ( x−5 ) 2 = 4 5 ( y+3 ),V:( 5,−3 );F:( 5,− 14 5 );d:y=− 16 5
( x−2 ) 2 =−2( y−5 ),V:( 2,5 );F:( 2, 9 2 );d:y= 11 2 ( x−2 ) 2 =−2( y−5 ),V:( 2,5 );F:( 2, 9 2 );d:y= 11 2
( y−1 ) 2 = 4 3 ( x−5 ),V:( 5,1 );F:( 16 3 ,1 );d:x= 14 3 ( y−1 ) 2 = 4 3 ( x−5 ),V:( 5,1 );F:( 16 3 ,1 );d:x= 14 3
x 2 =−16y x 2 =−16y
( y−2 ) 2 =4 2 ( x−2 ) ( y−2 ) 2 =4 2 ( x−2 )
( y+ 3 ) 2 =−4 2 ( x− 2 ) ( y+ 3 ) 2 =−4 2 ( x− 2 )
x 2 =y x 2 =y
( y−2 ) 2 = 1 4 ( x+2 ) ( y−2 ) 2 = 1 4 ( x+2 )
( y− 3 ) 2 =4 5 ( x+ 2 ) ( y− 3 ) 2 =4 5 ( x+ 2 )
y 2 =−8x y 2 =−8x
( y+1 ) 2 =12( x+3 ) ( y+1 ) 2 =12( x+3 )
( 0,1 ) ( 0,1 )
At the point 2.25 feet above the vertex.
0.5625 feet
x 2 =−125( y−20 ), x 2 =−125( y−20 ), height is 7.2 feet
2304 feet
12.4 Section Exercises
The xy xy term causes a rotation of the graph to occur.
The conic section is a hyperbola.
It gives the angle of rotation of the axes in order to eliminate the xy xy term.
AB=0, AB=0, parabola
AB=−4<0, AB=−4<0, hyperbola
AB=6>0, AB=6>0, ellipse
B 2 −4AC=0, B 2 −4AC=0, parabola
B 2 −4AC=0, B 2 −4AC=0, parabola
B 2 −4AC=−96<0, B 2 −4AC=−96<0, ellipse
7 x ′ 2 +9 y ′ 2 −4=0 7 x ′ 2 +9 y ′ 2 −4=0
3 x ′ 2 +2 x ′ y ′ −5 y ′ 2 +1=0 3 x ′ 2 +2 x ′ y ′ −5 y ′ 2 +1=0
θ= 60 ∘ ,11 x ′ 2 − y ′ 2 + 3 x ′ + y ′ −4=0 θ= 60 ∘ ,11 x ′ 2 − y ′ 2 + 3 x ′ + y ′ −4=0
θ= - 30 ∘ ,21 x ′ 2 +9 y ′ 2 +4 x ′ −4 3 y ′ −6=0 θ= - 30 ∘ ,21 x ′ 2 +9 y ′ 2 +4 x ′ −4 3 y ′ −6=0
θ≈ 36.9 ∘ ,125 x ′ 2 +6 x ′ −42 y ′ +10=0 θ≈ 36.9 ∘ ,125 x ′ 2 +6 x ′ −42 y ′ +10=0
θ= 45 ∘ ,3 x ′ 2 − y ′ 2 − 2 x ′ + 2 y ′ +1=0 θ= 45 ∘ ,3 x ′ 2 − y ′ 2 − 2 x ′ + 2 y ′ +1=0
2 2 ( x ′ + y ′ )= 1 2 ( x ′ − y ′ ) 2 2 2 ( x ′ + y ′ )= 1 2 ( x ′ − y ′ ) 2
( x ′ − y ′ ) 2 8 + ( x ′ + y ′ ) 2 2 =1 ( x ′ − y ′ ) 2 8 + ( x ′ + y ′ ) 2 2 =1
( x ′ + y ′ ) 2 2 − ( x ′ − y ′ ) 2 2 =1 ( x ′ + y ′ ) 2 2 − ( x ′ − y ′ ) 2 2 =1
3 2 x ′ − 1 2 y ′ = ( 1 2 x ′ + 3 2 y ′ −1 ) 2 3 2 x ′ − 1 2 y ′ = ( 1 2 x ′ + 3 2 y ′ −1 ) 2
θ= 45 ∘ θ= 45 ∘
θ= 60 ∘ θ= 60 ∘
θ≈ 36.9 ∘ θ≈ 36.9 ∘
−4 6 <k<4 6 −4 6 <k<4 6
k=2 k=2
12.5 Section Exercises
If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.
The directrix will be parallel to the polar axis.
One of the foci will be located at the origin.
Parabola with e=1 e=1 and directrix 3 4 3 4 units below the pole.
Hyperbola with e=2 e=2 and directrix 5 2 5 2 units above the pole.
Parabola with e=1 e=1 and directrix 3 10 3 10 units to the right of the pole.
Ellipse with e= 2 7 e= 2 7 and directrix 2 2 units to the right of the pole.
Hyperbola with e= 5 3 e= 5 3 and directrix 11 5 11 5 units above the pole.
Hyperbola with e= 8 7 e= 8 7 and directrix 7 8 7 8 units to the right of the pole.
25 x 2 +16 y 2 −12y−4=0 25 x 2 +16 y 2 −12y−4=0
21 x 2 −4 y 2 −30x+9=0 21 x 2 −4 y 2 −30x+9=0
64 y 2 =48x+9 64 y 2 =48x+9
96 y 2 −25 x 2 +110y+25=0 96 y 2 −25 x 2 +110y+25=0
3 x 2 +4 y 2 −2x−1=0 3 x 2 +4 y 2 −2x−1=0
5 x 2 +9 y 2 −24x−36=0 5 x 2 +9 y 2 −24x−36=0
r= 4 5+cosθ r= 4 5+cosθ
r= 4 1+2sinθ r= 4 1+2sinθ
r= 1 1+cosθ r= 1 1+cosθ
r= 7 8−28cosθ r= 7 8−28cosθ
r= 12 2+3sinθ r= 12 2+3sinθ
r= 15 4−3cosθ r= 15 4−3cosθ
r= 3 3−3cosθ r= 3 3−3cosθ
r=± 2 1+sinθcosθ r=± 2 1+sinθcosθ
r=± 2 4cosθ+3sinθ r=± 2 4cosθ+3sinθ
Review Exercises
x 2 5 2 + y 2 8 2 =1; x 2 5 2 + y 2 8 2 =1; center: ( 0,0 ); ( 0,0 ); vertices: ( 5,0 ),( −5,0 ),( 0,8 ),( 0,−8 ); ( 5,0 ),( −5,0 ),( 0,8 ),( 0,−8 ); foci: ( 0, 39 ),( 0,− 39 ) ( 0, 39 ),( 0,− 39 )
(x+3) 2 1 2 + (y−2) 2 3 2 =1(−3,2);(−2,2),(−4,2),(−3,5),(−3,−1);( −3,2+2 2 ),( −3,2−2 2 ) (x+3) 2 1 2 + (y−2) 2 3 2 =1(−3,2);(−2,2),(−4,2),(−3,5),(−3,−1);( −3,2+2 2 ),( −3,2−2 2 )
center: ( 0,0 ); ( 0,0 ); vertices: ( 6,0 ),( −6,0 ),( 0,3 ),( 0,−3 ); ( 6,0 ),( −6,0 ),( 0,3 ),( 0,−3 ); foci: ( 3 3 ,0 ),( −3 3 ,0 ) ( 3 3 ,0 ),( −3 3 ,0 )
center: ( −2,−2 ); ( −2,−2 ); vertices: ( 2,−2 ),( −6,−2 ),( −2,6 ),( −2,−10 ); ( 2,−2 ),( −6,−2 ),( −2,6 ),( −2,−10 ); foci: ( −2,−2+4 3 , ),( −2,−2−4 3 ) ( −2,−2+4 3 , ),( −2,−2−4 3 )
x 2 25 + y 2 16 =1 x 2 25 + y 2 16 =1
Approximately 35.71 feet
( y+1 ) 2 4 2 − ( x−4 ) 2 6 2 =1; ( y+1 ) 2 4 2 − ( x−4 ) 2 6 2 =1; center: ( 4,−1 ); ( 4,−1 ); vertices: ( 4,3 ),( 4,−5 ); ( 4,3 ),( 4,−5 ); foci: ( 4,−1+2 13 ),( 4,−1−2 13 ) ( 4,−1+2 13 ),( 4,−1−2 13 )
( x−2 ) 2 2 2 − ( y+3 ) 2 ( 2 3 ) 2 =1; ( x−2 ) 2 2 2 − ( y+3 ) 2 ( 2 3 ) 2 =1; center: ( 2,−3 ); ( 2,−3 ); vertices: ( 4,−3 ),( 0,−3 ); ( 4,−3 ),( 0,−3 ); foci: ( 6,−3 ),( −2,−3 ) ( 6,−3 ),( −2,−3 )
( x−5 ) 2 1 − ( y−7 ) 2 3 =1 ( x−5 ) 2 1 − ( y−7 ) 2 3 =1
( x+2 ) 2 = 1 2 ( y−1 ); ( x+2 ) 2 = 1 2 ( y−1 ); vertex: ( −2,1 ); ( −2,1 ); focus: ( −2, 9 8 ); ( −2, 9 8 ); directrix: y= 7 8 y= 7 8
( x+5 ) 2 =( y+2 ); ( x+5 ) 2 =( y+2 ); vertex: ( −5,−2 ); ( −5,−2 ); focus: ( −5,− 7 4 ); ( −5,− 7 4 ); directrix: y=− 9 4 y=− 9 4
( x−2 ) 2 =( 1 2 )( y−1 ) ( x−2 ) 2 =( 1 2 )( y−1 )
B 2 −4AC=0, B 2 −4AC=0, parabola
B 2 −4AC=−31<0, B 2 −4AC=−31<0, ellipse
θ= 45 ∘ , x ′ 2 +3 y ′ 2 −12=0 θ= 45 ∘ , x ′ 2 +3 y ′ 2 −12=0
θ= 45 ∘ θ= 45 ∘
Hyperbola with e=5 e=5 and directrix 2 2 units to the left of the pole.
Ellipse with e= 3 4 e= 3 4 and directrix 1 3 1 3 unit above the pole.
r= 3 1+cos θ r= 3 1+cos θ
Practice Test
x 2 3 2 + y 2 2 2 =1; x 2 3 2 + y 2 2 2 =1; center: ( 0,0 ); ( 0,0 ); vertices: ( 3,0 ),( –3,0 ),( 0,2 ),( 0,−2 ); ( 3,0 ),( –3,0 ),( 0,2 ),( 0,−2 ); foci: ( 5 ,0 ),( − 5 ,0 ) ( 5 ,0 ),( − 5 ,0 )
center: ( 3,2 ); ( 3,2 ); vertices: ( 11,2 ),( −5,2 ),( 3,8 ),( 3,−4 ); ( 11,2 ),( −5,2 ),( 3,8 ),( 3,−4 ); foci: ( 3+2 7 ,2 ),( 3−2 7 ,2 ) ( 3+2 7 ,2 ),( 3−2 7 ,2 )
( x−1 ) 2 36 + ( y−2 ) 2 27 =1 ( x−1 ) 2 36 + ( y−2 ) 2 27 =1
x 2 7 2 − y 2 9 2 =1; x 2 7 2 − y 2 9 2 =1; center: ( 0,0 ); ( 0,0 ); vertices ( 7,0 ),( −7,0 ); ( 7,0 ),( −7,0 ); foci: ( 130 ,0 ),( − 130 ,0 ); ( 130 ,0 ),( − 130 ,0 ); asymptotes: y=± 9 7 x y=± 9 7 x
center: ( 3,−3 ); ( 3,−3 ); vertices: ( 8,−3 ),( −2,−3 ); ( 8,−3 ),( −2,−3 ); foci: ( 3+ 26 ,−3 ),( 3− 26 ,−3 ); ( 3+ 26 ,−3 ),( 3− 26 ,−3 ); asymptotes: y=± 1 5 (x−3)−3 y=± 1 5 (x−3)−3
( y−3 ) 2 1 − ( x−1 ) 2 8 =1 ( y−3 ) 2 1 − ( x−1 ) 2 8 =1
( x−2 ) 2 = 1 3 ( y+1 ); ( x−2 ) 2 = 1 3 ( y+1 ); vertex: ( 2,−1 ); ( 2,−1 ); focus: ( 2,− 11 12 ); ( 2,− 11 12 ); directrix: y=− 13 12 y=− 13 12
Approximately 8.49 8.49 feet
parabola; θ≈ 63.4 ∘ θ≈ 63.4 ∘
x ′ 2 −4 x ′ +3 y ′ =0 x ′ 2 −4 x ′ +3 y ′ =0
Hyperbola with e= 3 2 , e= 3 2 , and directrix 5 6 5 6 units to the right of the pole.