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Soal No 17 halaman 66. Find a) the coordinates of the foci and vertices for hyperbola whose equations given, b) equation of the asymptotes. Sketch the curve. Penyelesaian :. Jelas Titik puncak adalah P( a,0) dan Q(-a,0) Jadi titik puncaknya adalah P(8,0) dan Q(-8,0). Jelas

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soal no 17 halaman 66
Soal No 17 halaman 66

Find a) the coordinates of the foci and vertices for hyperbola whose equations given, b) equation of the asymptotes.

Sketch the curve.

penyelesaian
Penyelesaian:

Jelas

Titikpuncakadalah P( a,0) dan Q(-a,0)

Jadititikpuncaknyaadalah P(8,0) dan Q(-8,0)

slide3

Jelas

Foci diperolehdari (c,0) dan (-c,0).

Jadifocinyaadalah

nomor 21 halaman 66
Nomor 21 Halaman 66

Sketch the two equation of each system on the same set of axes and specify the number of real number of the system.

slide7

Penyelesaian

Jelaspersamaan (1) adalahpersamaanlingkarankarena A=B denganpanjangjari-jari 3.

Jelaspersamaan (2) adalahpersamaanhiperbola, karenatanda A tidaksamadengantanda B dan

jelas persamaan 2
JelasPersamaan (2) :

Titikpuncak P(a,0) dan Q(-a,0).

Jadititikpuncaknya P(2,0) dan Q(-2,0).

Jelas foci diperolehdari (c,0) dan (-c,0) sehingga

Jadifocinyaadalah

slide10

Exercise 2.4 number 27

Find a standard equation of the hyperbola that has the foci of ellips 9x + 4y = 36 for vertices and the vertices of the ellips for foci.

slide11

Persamaan elips 9x + 4y = 36

Persamaan tersebut dapat dibentuk menjadi persamaan baku dari ellips dengan kedua ruas dikalikan , diperoleh:

Ingat : Ada 2 bentuk persamaan ellipsatau

slide12

Jelas

sehinggapersamaanberbentuk

dengan fokus pada sumbu Y.

Dari persaman tersebut diperoleh

dan

Nilai a menunjukkanabsiskoordinat titik puncak pada sumbu Y.

Jadi diperoleh koordinat titik puncaknya adalah

slide13

Untuk mencari fokus dapat dapat diperoleh dengan mensubstitusikan nilai a = 3 dan b = 2 dalam persamaan maka diperoleh:

slide14

Nilai c menunjukkan ordinatkoordinat titik fokus pada sumbu Y jadi fokusnya

Karena titik puncak elips merupakan titik fokus hiperbola dan titik fokus elips merupakan titik puncak hiperbola maka titik fokus hiperbola adalah dan

titik puncak hiperbola adalah

slide15

Mencaripersamanhiperbola

Substitusikan dan ke persamaan

, maka diperoleh

Jadipersamaanhiperbola

soal no 28 halaman 66
Soal No 28 Halaman 66

Find the length of the perpendicular segment from a focus of the hyperbola to one of the asymptotes.

Answer

From the equations, we get the foci of hyperbola are M(0,c) and N(0,-c)

So one of the focus is P

slide17

The asymptote of the equation is

So the coordinate of the asymptote (b,a)

The length of the perpendicular segment from a focus of hyperbola and one of the asymptote is

nomor 19 halaman 70
Nomor 19 Halaman 70

Name and sketch the graph of each equation

Solution :

Because of A and C same sign, B=0 so the name of graph from the equation is two parallel lines.

soal no 22 halaman 70
Soal No 22 Halaman 70

Graph the set of points contained in the graphs of

  • Both and
  • Either or
penyelesaian1
Penyelesaian

Jelasadalahpersamaanlingkarankarena A=B, denganpanjangjari-jari 6.

Jelasadalahpersamaan parabola.

Jika x= 0 maka y= -6

x=1 maka y= -5

x=-1 maka y= -5

x=2 maka y= -2

x=-2 maka y=-2

nomor 24 halaman 70
Nomor 24 Halaman 70

Graph each pair of conjugate hyperbola

Solution

persamaan 1
Persamaan 1

Jelas

Titikpuncakdarihiperboladiperolehdari P(0,a) and Q(0,-a)

Jelas

JadititikpuncaknyaadalahP(0, 3) and Q (0,-3)

persamaan 2
Persamaan 2

Jelas

TitikpuncakdarihiperboladiperolehdariA(a,0) dan B (-a,0)

Jelas

JadititikpuncaknyaadalahA(2,0) dan B(-2,0)

hal 70 no 25
Hal 70 no.25

Show that the foci of the conjugate hyperbolas

and lie on circle.

Solution

A hyperbola has the standard equation

From , we get

Foci of are A(0,c) and B(0,-c)

slide31

From , we also get

Foci of is M(0,c) and N(0,-c)

So for both hyperbolas and

And A=B, hence all foci are equidistant from a common point (the origin) and therefore lie on a circle

hal 66 no 29
Hal 66 no.29

By solving for y, obtain the equation

and argue that the graph

of the equation approaches the stright line graphs of as increases.

solution
Solution

As grows larger and larger,

approaches

0 and

Approaches 1, from which y approaches

Jelas

hal 66 no 30
Hal 66 no.30

By solving for x, obtain the equation

and argue that the graph

of the equation approaches the stright line graphs of as increases.

solution1
Solution

As grows larger and larger,

approaches

0 and

Approaches 1, from which x approaches

Jelas

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