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Jika garis  menyinggung elips , maka nilai ...

Pertanyaan

Jika garis y equals straight m x menyinggung elips open parentheses x minus 2 close parentheses squared over 4 plus open parentheses y plus 1 close parentheses squared over 2 equals 1, maka nilai 4 straight m equals...

Pembahasan Soal:

Diketahui bahwa garis y equals straight m x menyinggung elips open parentheses x minus 2 close parentheses squared over 4 plus open parentheses y plus 1 close parentheses squared over 2 equals 1. Untuk mencari nilai straight m, substitusikan y equals straight m x ke dalam persamaan elips.


table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses x minus 2 close parentheses squared over 4 plus open parentheses y plus 1 close parentheses squared over 2 end cell equals cell 1 space space space space space space space cross times 4 end cell row cell open parentheses x minus 2 close parentheses squared plus 2 open parentheses y plus 1 close parentheses squared end cell equals 4 row cell open parentheses x squared minus 4 x plus 4 close parentheses plus 2 open parentheses y squared plus 2 y plus 1 close parentheses end cell equals 4 row cell open parentheses x squared minus 4 x plus 4 close parentheses plus 2 open parentheses open parentheses straight m x close parentheses squared plus 2 open parentheses straight m x close parentheses plus 1 close parentheses end cell equals 4 row cell open parentheses x squared minus 4 x plus 4 close parentheses plus 2 open parentheses straight m squared x squared plus 2 straight m x plus 1 close parentheses end cell equals 4 row cell x squared minus 4 x plus 4 plus 2 straight m squared x squared plus 4 straight m x plus 2 end cell equals 4 row cell x squared plus 2 straight m squared x squared minus 4 x plus 4 straight m x plus 4 plus 2 minus 4 end cell equals 0 row cell open parentheses 2 straight m squared plus 1 close parentheses x squared plus open parentheses 4 straight m minus 4 close parentheses x plus 2 end cell equals 0 end table


Persamaan di atas merupakan persamaan kuadrat dengan straight a equals 2 straight m squared plus 1straight b equals 4 straight m minus 4 dan straight c equals 2. Karena garis y equals straight m x menyinggung elips, maka diskriminan dari persamaan kuadrat di atas sama dengan 0 open parentheses straight D equals 0 close parentheses.


table attributes columnalign right center left columnspacing 0px end attributes row straight D equals 0 row cell straight b squared minus 4 ac end cell equals 0 row cell open parentheses 4 straight m minus 4 close parentheses squared minus 4 open parentheses 2 straight m squared plus 1 close parentheses open parentheses 2 close parentheses end cell equals 0 row cell 16 straight m squared minus 32 straight m plus 16 minus 16 straight m squared minus 8 end cell equals 0 row cell negative 32 straight m plus 8 end cell equals 0 row cell negative 32 straight m end cell equals cell negative 8 end cell row straight m equals cell 1 fourth end cell end table


Maka,


table attributes columnalign right center left columnspacing 0px end attributes row cell 4 straight m end cell equals cell 4 times 1 fourth end cell row blank equals 1 end table


Jadi nilai 4 straight m equals 1.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 19 Mei 2021

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