10.E: Analytic Jiometri (Mazoezi)

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10.1: duaradufu

Katika sehemu hii, sisi kuchunguza sura ya chumba hiki na maombi yake halisi ya dunia, ikiwa ni pamoja na jinsi mbali mbali watu wawili katika Statuary Hall wanaweza kusimama na bado kusikia kila mmoja whisper.

Maneno

1) Eleza ellipse kwa suala la foci yake.

Jibu: Ellipse ni seti ya pointi zote katika ndege, jumla ya umbali wake kutoka pointi mbili fasta, inayoitwa foci, ni mara kwa mara.

2) Ambapo lazima foci ya ellipse uongo?

3) Ni kesi gani maalum ya ellipse tunayo wakati mhimili mkubwa na mdogo una urefu sawa?

Jibu: Kesi hii maalum itakuwa mduara.

4) Kwa kesi maalum iliyotajwa hapo juu, itakuwa nini kweli kuhusu foci ya ellipse hiyo?

5) Ni nini kinachoweza kusema juu ya ulinganifu wa grafu ya ellipse na kituo cha asili na foci kando ya$y$ -axis?

Jibu: Ni sawa kuhusu$x$ -axis,$y$ -axis, na asili.

Kialjebra

Kwa mazoezi 6-10, onyesha kama equations iliyotolewa inawakilisha ellipses. Ikiwa ndiyo, andika kwa fomu ya kawaida.

6)$2x^2 +y=4$

7)$4x^2 + 9y^2=36$

Jibu: ndiyo;$\dfrac{x^2}{3^2}+\dfrac{y^2}{2^2}=1$

8)$4x^2 - y^2=4$

9)$4x^2 + 9y^2=1$

Jibu: ndiyo;$\dfrac{x^2}{\left (\tfrac{1}{2} \right )^2}+\dfrac{y^2}{\left (\tfrac{1}{2} \right )^2}=1$

10)$4x^2-8x+9y^2-72y+112=0$

Kwa mazoezi 11-26, andika equation ya duaradufu katika fomu ya kawaida, na kutambua pointi za mwisho za shoka kuu na ndogo pamoja na foci.

11)$\dfrac{x^2}{4}+\dfrac{y^2}{49}=1$

Jibu: $\dfrac{x^2}{2^2}+\dfrac{y^2}{7^2}=1$; Endpoints ya mhimili kuu$(0,7)$ na$(0,-7)$. Mwisho wa mhimili mdogo$(2,0)$ na$(-2,0)$. Foci katika$(0, 3\sqrt{5})$,$(0, -3\sqrt{5})$.

12)$\dfrac{x^2}{100}+\dfrac{y^2}{64}=1$

13)$x^2 + 9y^2 = 1$

Jibu: $\dfrac{x^2}{(1)^2}+\dfrac{y^2}{\left (\tfrac{1}{3} \right )^2}=1$; Endpoints ya mhimili kuu$(1,0)$ na$(-1,0)$. Mwisho wa mhimili mdogo$\left (0, \dfrac{1}{3} \right )$,$\left (0, -\dfrac{1}{3} \right )$. Foci katika$\left (\dfrac{2\sqrt{2}}{3}, 0 \right )$,$\left (-\dfrac{2\sqrt{2}}{3}, 0 \right )$.

14)$4x^2 + 16y^2 = 1$

15)$\dfrac{(x-2)^2}{49}+\dfrac{(y-4)^2}{25}=1$

Jibu: $\dfrac{(x-2)^2}{7^2}+\dfrac{(y-4)^2}{5^2}=1$; Endpoints ya mhimili mkubwa$(9,4)$,$(-5,4)$. Mwisho wa mhimili mdogo$(2,9)$,$(2,-1)$. Foci katika$(2+2\sqrt{6}, 4)$,$(2-2\sqrt{6}, 4)$

16)$\dfrac{(x-2)^2}{81}+\dfrac{(y+1)^2}{16}=1$

17)$\dfrac{(x+5)^2}{4}+\dfrac{(y-7)^2}{9}=1$

Jibu: $\dfrac{(x+5)^2}{2^2}+\dfrac{(y-7)^2}{3^2}=1$; Endpoints ya mhimili mkubwa$(-5,10)$,$(-5,4)$. Mwisho wa mhimili mdogo$(-3,7)$,$(-7,7)$. Foci katika$(-5, 7+\sqrt{5})$,$(-5, 7-\sqrt{5})$

18)$\dfrac{(x-7)^2}{49}+\dfrac{(y-7)^2}{49}=1$

19)$4x^2-8x+9y^2-72y+112=0$

Jibu: $\dfrac{(x-1)^2}{3^2}+\dfrac{(y-4)^2}{2^2}=1$; Endpoints ya mhimili mkubwa$(4,4)$,$(-2,4)$. Mwisho wa mhimili mdogo$(1,6)$,$(1,2)$. Foci katika$(1+\sqrt{5}, 4)$,$(1-\sqrt{5}, 4)$

20)$9x^2-54x+9y^2-54y+81=0$

21)$4x^2-24x+36y^2-360y+864=0$

Jibu: $\dfrac{(x-3)^2}{(3\sqrt{2})^2}+\dfrac{(y-5)^2}{\sqrt{2}^2}=1$; Endpoints ya mhimili mkubwa$(3+3\sqrt{2}, 5)$,$(3-3\sqrt{2}, 5)$. Mwisho wa mhimili mdogo$(3, 5+\sqrt{2})$,$(3, 5-\sqrt{2})$. Foci katika$(7,5)$,$(-1,5)$

22)$4x^2+24x+16y^2-128y+228=0$

23)$4x^2+40x+25y^2-100y+100=0$

Jibu: $\dfrac{(x+5)^2}{(5)^2}+\dfrac{(y-2)^2}{(2)^2}=1$; Endpoints ya mhimili mkubwa$(0,2)$,$(-10,2)$. Mwisho wa mhimili mdogo$(-5,4)$,$(-5,0)$. Foci katika$(-5+\sqrt{21}, 2)$,$(-5-\sqrt{21}, 2)$

24)$x^2+2x+100y^2-1000y+2401=0$

25)$4x^2+24x+25y^2+200y+336=0$

Jibu: $\dfrac{(x+3)^2}{(5)^2}+\dfrac{(y+4)^2}{(2)^2}=1$; Endpoints ya mhimili mkubwa$(2,-4)$,$(-8,-4)$. Mwisho wa mhimili mdogo$(-3,-2)$,$(-3,-6)$. Foci katika$(-3+\sqrt{21}, -4)$,$(-3-\sqrt{21}, -4)$

26)$9x^2+72x+16y^2+16y+4=0$

Kwa mazoezi 27-31, tafuta foci kwa ellipses iliyotolewa.

27)$\dfrac{(x+3)^2}{25}+\dfrac{(y+1)^2}{36}=1$

Jibu: Foci$(-3, -1+\sqrt{11})$,$(-3, -1-\sqrt{11})$

28)$\dfrac{(x+1)^2}{100}+\dfrac{(y-2)^2}{4}=1$

29)$x^2+y^2=1$

Jibu: Focus$(0,0)$

30)$x^2+4y^2+4x+8y=1$

31)$10x^2+y^2+200x=0$

Jibu: Foci$(-10,30)$,$(-10,-30)$

Graphic

Kwa mazoezi 32-45, graph ellipses iliyotolewa, akibainisha kituo, vertices, na foci.

32)$\dfrac{x^2}{25}+\dfrac{y^2}{36}=1$

33)$\dfrac{x^2}{16}+\dfrac{y^2}{9}=1$

Jibu

Kituo cha$(0,0)$, vipeo$(4,0)$$(-4,0)$,$(0,3)$,$(0,-3)$. Foci$(\sqrt{7},0)$,$(-\sqrt{7},0)$

$Ex 10.1.33.png$

34)$4x^2+9y^2=1$

35)$81x^2+49y^2=1$

Jibu

Kituo cha$(0,0)$, vipeo$\left ( \dfrac{1}{9}, 0 \right )$$\left ( -\dfrac{1}{9}, 0 \right )$,$\left ( 0, \dfrac{1}{7} \right )$,$\left ( 0, \dfrac{1}{7} \right )$. Foci$\left ( 0, \dfrac{4\sqrt{2}}{63} \right )$,$\left ( 0, -\dfrac{4\sqrt{2}}{63} \right )$

$Ex 10.1.35.png$

36)$\dfrac{(x-2)^2}{64}+\dfrac{(y-4)^2}{16}=1$

37)$\dfrac{(x+3)^2}{9}+\dfrac{(y-3)^2}{9}=1$

Jibu

Kituo cha$(-3,3)$, vipeo$(0,3)$$(-6,3)$,$(-3,0)$,$(-3,6)$. Focus$(-3,3)$

Kumbuka kwamba duaradufu hii ni mduara. Mzunguko una lengo moja tu, linalofanana na kituo.

$Ex 10.1.37.png$

38)$\dfrac{x^2}{2}+\dfrac{(y+1)^2}{5}=1$

39)$4x^2-8x+16y^2-32y-44=0$

Jibu

Kituo cha$(1,1)$, vipeo$(5,1)$$(-3,1)$,$(1,3)$,$(1,-1)$. Foci$(1,1+4\sqrt{3})$,$(1,1-4\sqrt{3})$

$Ex 10.1.39.png$

40)$x^2-8x+25y^2-100y+91=0$

41)$x^2+8x+4y^2-40y+112=0$

Jibu

Kituo cha$(-4,5)$, vipeo$(-2,5)$$(-6,4)$,$(-4,6)$,$(-4,4)$. Foci$(-4+\sqrt{3}, 5)$,$(-4-4\sqrt{3}, 5)$

$Ex 10.1.41.png$

42)$64x^2+128x+9y^2-72y-368=0$

43)$16x^2+64x+4y^2-8y+4=0$

Jibu

Kituo cha$(-2,1)$, vipeo$(0,1)$$(-4,1)$,$(-2,5)$,$(-2,-3)$. Foci$(-2,1+2\sqrt{3})$,$(-2,1-2\sqrt{3})$

$Ex 10.1.43.png$

44)$100x^2+1000x+y^2-10y+2425=0$

45)$4x^2+16x+4y^2+16y+16=0$

Jibu

Kituo cha$(-2,2)$, vipeo$(0,-2)$$(-4,-2)$,$(-2,0)$,$(-2,-4)$. Focus$(-2,-2)$

$Ex 10.1.45.png$

Kwa mazoezi 46-51, tumia taarifa iliyotolewa kuhusu grafu ya kila duaradufu ili kuamua usawa wake.

46) Kituo cha asili, ulinganifu kwa heshima na$x$ - na$y$ -axes, kuzingatia$(4,0)$ na kumweka kwenye grafu$(0,3)$.

47) Kituo cha asili, ulinganifu kwa heshima na$x$ - na$y$ -axes, kuzingatia$(0,-2)$ na kumweka kwenye grafu$(5,0)$.

Jibu: $\dfrac{x^2}{25}+\dfrac{y^2}{29}=1$

48) Kituo cha asili, ulinganifu kwa heshima na$x$ - na$y$ -axes, kuzingatia$(3,0)$, na mhimili mkubwa ni mara mbili kwa muda mrefu kama mhimili mdogo.

49) Kituo$(4,2)$; vertex$(9,2)$; lengo moja:$(4+2\sqrt{6}, 2)$

Jibu: $\dfrac{(x-4)^2}{25}+\dfrac{(y-2)^2}{1}=1$

50) Kituo$(3,5)$; vertex$(3,11)$; lengo moja:$(3, 5+4\sqrt{2})$

51) Kituo$(-3,4)$; vertex$(1,4)$; lengo moja:$(-3+2\sqrt{3}, 4)$

Jibu: $\dfrac{(x+3)^2}{16}+\dfrac{(y-4)^2}{4}=1$

Kwa mazoezi 52-56, kutokana na grafu ya ellipse, tambua usawa wake.

52)

$Ex 10.1.52.png$

53)

$Ex 10.1.53.png$

Jibu: $\dfrac{x^2}{81}+\dfrac{y^2}{9}=1$

54)

$Ex 10.1.54.png$

55)

$Ex 10.1.55.png$

Jibu: $\dfrac{(x+2)^2}{4}+\dfrac{(y-2)^2}{9}=1$

56)

$Ex 10.1.56.png$

Upanuzi

Kwa mazoezi 57-61, tafuta eneo la ellipse. Eneo la ellipse linatolewa na formula$\text {Area}=a\cdot b\cdot \pi$

57)$\dfrac{(x-3)^2}{9}+\dfrac{(y-3)^2}{16}=1$

Jibu: $\text {Area} = 12\pi $vitengo vya mraba

58)$\dfrac{(x+6)^2}{16}+\dfrac{(y-6)^2}{36}=1$

59)$\dfrac{(x+1)^2}{4}+\dfrac{(y-2)^2}{5}=1$

Jibu: $\text {Area} = 2\sqrt{5} \pi $vitengo vya mraba

60)$4x^2-8x+9y^2-72y+112=0$

61)$9x^2-54x+9y^2-54y+81=0$

Jibu: $\text {Area} = 9\pi $vitengo vya mraba

Real-World Matumizi

62) Kupata equation ya duaradufu kwamba tu fit ndani ya sanduku yaani$8$ vitengo pana na$4$ vitengo juu.

63) Pata equation ya duaradufu ambayo inafaa tu ndani ya sanduku ambayo ni mara nne pana kama ilivyo juu. Express katika suala la$h$, urefu.

Jibu: $\dfrac{x^2}{4h^2}+\dfrac{y^2}{\tfrac{1}{4}h^2}=1$

64) Arch ina sura ya nusu-ellipse (nusu ya juu ya ellipse). Arch ina urefu wa$8$ miguu na muda wa$20$ miguu. Pata equation kwa duaradufu, na uitumie ili kupata urefu kwa$0.01$ mguu wa karibu wa arch umbali wa$4$ miguu kutoka katikati.

65) Arch ina sura ya nusu-ellipse. Arch ina urefu wa$12$ miguu na muda wa$40$ miguu. Kupata equation kwa duaradufu, na kutumia kwamba kupata umbali kutoka katikati ya hatua ambayo urefu ni$6$ miguu. Pande zote hadi karibu na mia moja.

Jibu: $\dfrac{x^2}{400}+\dfrac{y^2}{144}=1$. Umbali =$17.32$ miguu

66) Daraja ni kujengwa kwa sura ya arch nusu-elliptical na ni kuwa na muda wa$120$ miguu. Urefu wa arch umbali wa$40$ miguu kutoka katikati ni kuwa$8$ miguu. Pata urefu wa arch katikati yake.

67) Mtu katika nyumba ya sanaa ya whispering amesimama kwenye mtazamo mmoja wa duaradufu anaweza kuongea na kusikilizwa na mtu amesimama kwenye mwelekeo mwingine kwa sababu mawimbi yote ya sauti ambayo yanafikia dari yanajitokeza kwa mtu mwingine. Ikiwa nyumba ya sanaa ya whispering ina urefu wa$120$$30$ miguu, na foci iko miguu kutoka katikati, pata urefu wa dari katikati.

Jibu: Takriban$51.96$ miguu

68) Mtu amesimama$8$ miguu kutoka ukuta wa karibu katika nyumba ya sanaa ya whispering. Kama mtu huyo ni katika lengo moja, na lengo nyingine ni$80$ miguu mbali, ni nini urefu na urefu katika kituo cha nyumba ya sanaa?

10.2: Hyperbola

Katika jiometri ya uchambuzi, hyperbola ni sehemu ya conic inayoundwa na kuingiliana koni ya mviringo ya mviringo na ndege kwa pembe kama vile nusu zote za koni zimeunganishwa. Mfululizo huu hutoa curves mbili tofauti zisizo na mipaka ambazo ni picha za kioo za kila mmoja.