Tentukan persamaan garis singgung pada lingkaran dari

Persamaan garis singgung lingkaran adalah y = -x + 2 ± √18

Pembahasan

Untuk menentukan persamaan garis singgung pada lingkaran dengan gradien yang diketahui, kita dapat menggunakan rumus garis singgung lingkaran $(x-h)^2 + (y-k)^2 = r^2$ dengan gradien $m$, yaitu $y-k = m(x-h) \\\pm r \\\sqrt{m^2+1}$.

Terlebih dahulu, kita perlu mengubah persamaan lingkaran $x^2+y^2-4x-2=0$ ke dalam bentuk standar $(x-h)^2 + (y-k)^2 = r^2$.

$x^2 - 4x + y^2 = 2$
$(x^2 - 4x + 4) + y^2 = 2 + 4$
$(x-2)^2 + y^2 = 6$

Dari persamaan standar tersebut, kita dapat mengidentifikasi bahwa pusat lingkaran $(h, k)$ adalah $(2, 0)$ dan jari-jarinya $r = \\\sqrt{6}$. Gradien yang diketahui adalah $m = -1$.

Sekarang, kita masukkan nilai-nilai ini ke dalam rumus garis singgung:
$y - 0 = -1(x - 2) \\\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\\]\" ,