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

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

Sketch the curve.


Penyelesaian:

Jelas

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

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


Jelas

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

Jadifocinyaadalah


Persamaanasimtotdarihyperbola

Dapatdiperolehdari


Gambargrafiknya


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.


Penyelesaian

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

Jelaspersamaan (2) adalahpersamaanhiperbola, karenatanda A tidaksamadengantanda B dan


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


Gambargrafiknya


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.


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


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


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


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


Mencaripersamanhiperbola

Substitusikan danke persamaan

, maka diperoleh

Jadipersamaanhiperbola


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


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

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


Soal No 22 Halaman 70

Graph the set of points contained in the graphs of

  • Both and

  • Either or


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 b


Nomor 24 Halaman 70

Graph each pair of conjugate hyperbola

Solution


Persamaan 1

Jelas

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

Jelas

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


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

Jelas

Jadifocinyaadalah


Persamaan 2

Jelas

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

Jelas

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


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

Jelas

Jadifocinyaadalah


Gambargrafiknya


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)


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

By solvingfor y, obtain the equation

and argue that the graph

of the equation approaches the stright line graphs ofasincreases.


Solution

Asgrows larger and larger,

approaches

0 and

Approaches 1, from which y approaches

Jelas


Hal 66 no.30

By solvingfor x, obtain the equation

and argue that the graph

of the equation approaches the stright line graphs ofasincreases.


Solution

Asgrows larger and larger,

approaches

0 and

Approaches 1, from which x approaches

Jelas


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