Tentukan persamaan garis singgung pada suatu lingkaran yang ditarik dari titik diluar lingkaran.

Pertanyaan

Tentukan persamaan garis singgung pada suatu lingkaran $begin mathsize 14px style x squared plus y squared plus 2 x minus 19 equals 0 end style$ yang ditarik dari titik $begin mathsize 14px style T left parenthesis 1 comma 6 right parenthesis end style$ diluar lingkaran.

Pembahasan Soal:

Misalkan persamaan garis singgungnya adalah :

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m left parenthesis x minus x subscript 1 right parenthesis end cell row cell y minus 6 end cell equals cell m left parenthesis x minus 1 right parenthesis end cell row y equals cell m x minus m plus 6 end cell end table end style$

Lalu substitusi persamaan garis singgung tersebut ke persamaan lingkaran soal.

$table attributes columnalign right center left columnspacing 0px end attributes row cell size 14px x to the power of size 14px 2 size 14px plus size 14px left parenthesis size 14px m size 14px x size 14px minus size 14px m size 14px plus size 14px 6 size 14px right parenthesis to the power of size 14px 2 size 14px plus size 14px 2 size 14px x size 14px minus size 14px 19 end cell size 14px equals size 14px 0 row cell size 14px x to the power of size 14px 2 size 14px plus size 14px m to the power of size 14px 2 size 14px x to the power of size 14px 2 size 14px plus size 14px m to the power of size 14px 2 size 14px plus size 14px 36 size 14px minus size 14px 2 size 14px m to the power of size 14px 2 size 14px x size 14px plus size 14px 12 size 14px m size 14px x size 14px minus size 14px 12 size 14px m size 14px plus size 14px 2 size 14px x size 14px minus size 14px 19 end cell size 14px equals size 14px 0 row cell size 14px left parenthesis size 14px 1 size 14px plus size 14px m to the power of size 14px 2 size 14px right parenthesis size 14px x to the power of size 14px 2 size 14px plus size 14px left parenthesis size 14px minus size 14px 2 size 14px m to the power of size 14px 2 size 14px plus size 14px 12 size 14px m size 14px plus size 14px 2 size 14px right parenthesis size 14px x size 14px plus size 14px m to the power of size 14px 2 size 14px minus size 14px 12 size 14px m size 14px plus size 14px 17 end cell size 14px equals size 14px 0 end table$

Agar garis menyinggung lingkaran, maka nilai D harus nol.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row D equals cell b squared minus 4 a c equals 0 end cell row D equals cell left parenthesis negative 2 m squared plus 12 m plus 2 right parenthesis squared minus 4 left parenthesis 1 plus m to the power of 2 end exponent right parenthesis left parenthesis m squared minus 12 m plus 17 right parenthesis equals 0 end cell row D equals cell up diagonal strike 4 m to the power of 4 end strike plus 144 m squared plus 4 minus up diagonal strike 48 m cubed end strike minus 8 m squared plus 48 m minus 4 m squared plus 48 m minus 68 minus up diagonal strike 4 m to the power of 4 end strike plus up diagonal strike 48 m cubed end strike minus 68 m squared equals 0 end cell row D equals cell 64 m squared plus 96 m minus 64 equals 0 end cell end table end style$

Diiperoleh $begin mathsize 14px style m subscript 1 equals negative 2 end style$ dan $begin mathsize 14px style m subscript 2 equals 1 half end style$ .

Maka, persamaan garis singgungnya adalah :

$begin mathsize 14px style y equals negative 2 x plus 8 end style$ dan $begin mathsize 14px style y equals 1 half x plus 11 over 2 end style$ .

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 02 Maret 2021

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