Soal no 17 halaman 66
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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

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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)


Soal no 17 halaman 66

Jelas

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

Jadifocinyaadalah


Soal no 17 halaman 66

Persamaanasimtotdarihyperbola

Dapatdiperolehdari


Gambar grafiknya

Gambargrafiknya


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.


Soal no 17 halaman 66

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


Gambar grafiknya1

Gambargrafiknya


Soal no 17 halaman 66

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.


Soal no 17 halaman 66

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


Soal no 17 halaman 66

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


Soal no 17 halaman 66

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


Soal no 17 halaman 66

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


Soal no 17 halaman 66

Mencaripersamanhiperbola

Substitusikan danke 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


Soal no 17 halaman 66

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.


Sketch of the graph

Sketch of the Graph


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


The graph of 22 a

The graph of 22 a


The graph of 22 b

The graph of 22 b


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)


Soal no 17 halaman 66

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

Jelas

Jadifocinyaadalah


Persamaan 2

Persamaan 2

Jelas

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

Jelas

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


Soal no 17 halaman 66

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

Jelas

Jadifocinyaadalah


Gambar grafiknya2

Gambargrafiknya


Hal 70 no 25

Hal 70 no.25

Show that the foci of the conjugate hyperbolas

andlie on circle.

Solution

A hyperbola has the standard equation

From , we get

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


Soal no 17 halaman 66

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 solvingfor y, obtain the equation

and argue that the graph

of the equation approaches the stright line graphs ofasincreases.


Solution

Solution

Asgrows larger and larger,

approaches

0 and

Approaches 1, from which y approaches

Jelas


Hal 66 no 30

Hal 66 no.30

By solvingfor x, obtain the equation

and argue that the graph

of the equation approaches the stright line graphs ofasincreases.


Solution1

Solution

Asgrows larger and larger,

approaches

0 and

Approaches 1, from which x approaches

Jelas


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