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Persamaan lingkaran yang melalui titik , , dan  adalah ....

Pertanyaan

Persamaan lingkaran yang melalui titik open parentheses 1 comma space 5 close parentheses, open parentheses negative 6 comma space 4 close parentheses, dan open parentheses negative 5 comma space minus 3 close parentheses adalah ....

Pembahasan Soal:

Dengan substitusi masing-masing titik ke bentuk umum persamaan lingkaran x squared plus y squared plus A x plus B y plus C equals 0, diperoleh sistem persamaan linear tiga variabel (SPLTV) sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 1 comma space 5 close parentheses colon end cell row cell 1 squared plus 5 squared plus A times 1 plus B times 5 plus C end cell equals 0 row cell 1 plus 25 plus A plus 5 B plus C end cell equals 0 row cell A plus 5 B plus C end cell equals cell negative 26 space... space left parenthesis 1 right parenthesis end cell row blank blank blank row blank blank cell open parentheses negative 6 comma space 4 close parentheses colon end cell row cell open parentheses negative 6 close parentheses squared plus 4 squared plus A times open parentheses negative 6 close parentheses plus B times 4 plus C end cell equals 0 row cell 36 plus 16 minus 6 A plus 4 B plus C end cell equals 0 row cell negative 6 A plus 4 B plus C end cell equals cell negative 52 space... space left parenthesis 2 right parenthesis end cell row blank blank blank row blank blank cell open parentheses negative 5 comma space minus 3 close parentheses colon end cell row cell open parentheses negative 5 close parentheses squared plus open parentheses negative 3 close parentheses squared plus A times open parentheses negative 5 close parentheses plus B times open parentheses negative 3 close parentheses plus C end cell equals 0 row cell 25 plus 9 minus 5 A minus 3 B plus C end cell equals 0 row cell negative 5 A minus 3 B plus C end cell equals cell negative 34 space... space left parenthesis 3 right parenthesis end cell end table end style 

Dalam bentuk matriks, SPLTV tersebut dapat dinyatakan sebagai berikut.

open parentheses table row 1 5 1 row cell negative 6 end cell 4 1 row cell negative 5 end cell cell negative 3 end cell 1 end table close parentheses open parentheses table row A row B row C end table close parentheses equals open parentheses table row cell negative 26 end cell row cell negative 52 end cell row cell negative 34 end cell end table close parentheses 

Dengan menerapkan rumus determinan matriks ordo 3 cross times 3, diperoleh:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row D equals cell open vertical bar table row 1 5 1 row cell negative 6 end cell 4 1 row cell negative 5 end cell cell negative 3 end cell 1 end table close vertical bar table row 1 5 row cell negative 6 end cell 4 row cell negative 5 end cell cell negative 3 end cell end table end cell row blank equals cell open parentheses 4 minus 25 plus 18 close parentheses minus open parentheses negative 20 minus 3 minus 30 close parentheses end cell row blank equals cell open parentheses negative 3 close parentheses minus open parentheses negative 53 close parentheses end cell row blank equals 50 row cell D subscript A end cell equals cell open vertical bar table row cell negative 26 end cell 5 1 row cell negative 52 end cell 4 1 row cell negative 34 end cell cell negative 3 end cell 1 end table close vertical bar table row cell negative 26 end cell 5 row cell negative 52 end cell 4 row cell negative 34 end cell cell negative 3 end cell end table end cell row blank equals cell open parentheses negative 104 minus 170 plus 156 close parentheses minus open parentheses negative 136 plus 78 minus 260 close parentheses end cell row blank equals cell open parentheses negative 118 close parentheses minus open parentheses negative 318 close parentheses end cell row blank equals 200 row cell D subscript B end cell equals cell open vertical bar table row 1 cell negative 26 end cell 1 row cell negative 6 end cell cell negative 52 end cell 1 row cell negative 5 end cell cell negative 34 end cell 1 end table close vertical bar table row 1 cell negative 26 end cell row cell negative 6 end cell cell negative 52 end cell row cell negative 5 end cell cell negative 34 end cell end table end cell row blank equals cell open parentheses negative 52 plus 130 plus 204 close parentheses minus open parentheses 260 minus 34 plus 156 close parentheses end cell row blank equals cell open parentheses 282 close parentheses minus open parentheses 382 close parentheses end cell row blank equals cell negative 100 end cell row cell D subscript C end cell equals cell open vertical bar table row 1 5 cell negative 26 end cell row cell negative 6 end cell 4 cell negative 52 end cell row cell negative 5 end cell cell negative 3 end cell cell negative 34 end cell end table close vertical bar table row 1 5 row cell negative 6 end cell 4 row cell negative 5 end cell cell negative 3 end cell end table end cell row blank equals cell open parentheses negative 136 plus 1.300 minus 468 close parentheses minus open parentheses 520 plus 156 plus 1.020 close parentheses end cell row blank equals cell open parentheses 696 close parentheses minus open parentheses 1.696 close parentheses end cell row blank equals cell negative 1.000 end cell end table end style 

Nilai A, B, dan C dapat ditentukan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell D subscript A over D equals 200 over 50 equals 4 end cell row B equals cell D subscript B over D equals fraction numerator negative 100 over denominator 50 end fraction equals negative 2 end cell row C equals cell D subscript C over D equals fraction numerator negative 1.000 over denominator 50 end fraction equals negative 20 end cell end table 

Jadi, persamaan lingkaran tersebut adalah x squared plus y squared plus 4 x minus 2 y minus 20 equals 0.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 26 April 2021

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