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Lukislah DHP dari setiap SPtKKDV di bawah ini. 6. ⎩⎨⎧​4x2+9y2≥36y≥x2+6x+5y<3x+15​

Pertanyaan

Lukislah DHP dari setiap SPtKKDV di bawah ini.

6. open curly brackets table attributes columnalign left end attributes row cell 4 x squared plus 9 y squared greater or equal than 36 end cell row cell y greater or equal than x squared plus 6 x plus 5 end cell row cell y less than 3 x plus 15 end cell end table close   

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

Cari DHP dari 4 x squared plus 9 y squared greater or equal than 36 terlebih dahulu

Kurva pembatas dari 4 x squared plus 9 y squared greater or equal than 36 adalah 4 x squared plus 9 y squared equals 36  

Ubah bentuk persamaan tersebut menjadi seperti berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 x squared plus 9 y squared end cell equals 36 row cell fraction numerator 4 x squared plus 9 y squared over denominator 36 end fraction end cell equals 1 row cell fraction numerator 4 x squared over denominator 36 end fraction plus fraction numerator 9 y squared over denominator 36 end fraction end cell equals 1 row cell x squared over 9 plus y squared over 4 end cell equals 1 row cell x squared over 3 squared plus y squared over 2 squared end cell equals 1 end table  

x squared over a squared plus y squared over b squared equals 1 dengan a greater than b merupakan bentuk persamaan elips horizontal dengan titik pusat left parenthesis 0 comma space 0 right parenthesis dengan left parenthesis a comma space 0 right parenthesis dan left parenthesis negative a comma space 0 right parenthesis adalah titik potong sumbu x dan left parenthesis 0 comma space b right parenthesis dan left parenthesis 0 comma space minus b right parenthesis adalah titik potong sumbu y maka dari persamaan x squared over 3 squared plus y squared over 2 squared equals 1 diketahui bahwa

table attributes columnalign right center left columnspacing 0px end attributes row pusat equals cell left parenthesis 0 comma space 0 right parenthesis end cell row cell tipot space sb space x end cell equals cell left parenthesis 3 comma space 0 right parenthesis space dan space left parenthesis negative 3 comma space 0 right parenthesis end cell row cell tipot space sb space y end cell equals cell left parenthesis 0 comma space 2 right parenthesis space dan space left parenthesis 0 comma space minus 2 right parenthesis end cell end table  

Untuk mengetahui letak DHP lakukan uji titik

untuk left parenthesis 0 comma space 0 right parenthesis 

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 x squared plus 9 y squared end cell greater or equal than 36 row cell 4 left parenthesis 0 right parenthesis squared plus 9 left parenthesis 0 right parenthesis squared end cell greater or equal than 36 row 0 greater or equal than cell 36 space left parenthesis salah right parenthesis end cell end table  

karena titik left parenthesis 0 comma space 0 right parenthesis salah maka daerah yang terdapat titik bukan merupakan DHP dari 4 x squared plus 9 y squared greater or equal than 36 .

Selanjutnya cari DHP dari y greater or equal than x squared plus 6 x plus 5 

Kurva pembatas dari y greater or equal than x squared plus 6 x plus 5 adalah y equals x squared plus 6 x plus 5   

Berdasarkan bentuk umum fungsi kuadrat y equals a x squared plus b x plus c maka diketahui

a equals 1 b equals 6 c equals 5 

Cari titik puncak dari bentuk persamaan y equals a x squared plus b x plus c menggunakan rumus left parenthesis x subscript p comma space y subscript p right parenthesis equals left parenthesis negative fraction numerator b over denominator 2 a end fraction comma space minus fraction numerator b squared minus 4 a c over denominator 4 a end fraction right parenthesis 

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis x subscript p comma space y subscript p right parenthesis end cell equals cell left parenthesis negative fraction numerator b over denominator 2 a end fraction comma space minus fraction numerator b squared minus 4 a c over denominator 4 a end fraction right parenthesis end cell row blank equals cell left parenthesis negative fraction numerator 6 over denominator 2 left parenthesis 1 right parenthesis end fraction comma space minus fraction numerator 6 squared minus 4 left parenthesis 1 right parenthesis left parenthesis 5 right parenthesis over denominator 4 left parenthesis 1 right parenthesis end fraction right parenthesis end cell row blank equals cell left parenthesis negative 3 comma space minus fraction numerator 36 minus 20 over denominator 4 end fraction right parenthesis end cell row blank equals cell left parenthesis negative 3 comma space minus 16 over 4 right parenthesis end cell row blank equals cell left parenthesis negative 3 comma space minus 4 right parenthesis end cell end table 

Cari titik potong sumbu xy equals 0  

       table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x squared plus 6 x plus 5 end cell row 0 equals cell x squared plus 6 x plus 5 end cell row 0 equals cell left parenthesis x plus 5 right parenthesis left parenthesis x plus 1 right parenthesis end cell row blank blank cell x equals negative 5 logical or x equals negative 1 end cell row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis negative 5 comma space 0 right parenthesis end cell row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis negative 1 comma space 0 right parenthesis end cell end table 

Cari titik potong sumbu xy equals 0 

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x squared plus 6 x plus 5 end cell row y equals cell 0 squared plus 6 left parenthesis 0 right parenthesis plus 5 end cell row y equals 5 row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis 0 comma space 5 right parenthesis end cell end table 

Untuk mengetahui letak DHP lakukan uji titik

untuk left parenthesis 0 comma space 0 right parenthesis    

y greater or equal than x squared plus 6 x plus 5 0 greater or equal than 0 squared plus 6 left parenthesis 0 right parenthesis plus 5 0 greater or equal than 5 space left parenthesis salah right parenthesis   

karena titik left parenthesis 0 comma space 0 right parenthesis salah maka daerah yang terdapat titik left parenthesis 0 comma space 0 right parenthesis bukan merupakan DHP dari y greater or equal than x squared plus 6 x plus 5  

Selanjutnya cari DHP dari y less than 3 x plus 15 

Garis pembatas dari y less than 3 x plus 15 adalah y equals 3 x plus 15 

Cari titik potong sumbu x space dan space y dari y equals 3 x plus 15

untuk x equals 0 maka y equals 15 comma space left parenthesis x comma space y right parenthesis equals left parenthesis 0 comma space 15 right parenthesis 
untuk y equals 0 maka x equals negative 5 comma space left parenthesis x comma space y right parenthesis equals left parenthesis negative 5 comma space 0 right parenthesis 

Untuk mengetahui letak DHP lakukan uji titik

untuk left parenthesis 0 comma space 0 right parenthesis 

y less than 3 x plus 15 0 less than 3 left parenthesis 0 right parenthesis plus 15 0 less than 15 space left parenthesis benar right parenthesis 

karena titik left parenthesis 0 comma space 0 right parenthesis benar maka daerah yang terdapat titik left parenthesis 0 comma space 0 right parenthesis merupakan DHP dari y less than 3 x plus 15.

DHP dari open curly brackets table attributes columnalign left end attributes row cell 4 x squared plus 9 y squared greater or equal than 36 end cell row cell y greater or equal than x squared plus 6 x plus 5 end cell row cell y less than 3 x plus 15 end cell end table close adalah irisan dari DHP 4 x squared plus 9 y squared greater or equal than 36, DHP y greater or equal than x squared plus 6 x plus 5 dan DHP y less than 3 x plus 15.

Jadi, DHP dari open curly brackets table attributes columnalign left end attributes row cell 4 x squared plus 9 y squared greater or equal than 36 end cell row cell y greater or equal than x squared plus 6 x plus 5 end cell row cell y less than 3 x plus 15 end cell end table closeyaitu

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