# tangent lines of a circle

find tangent lines through a point (a,b) out of a circle x^2+y^2=r^2.

tangent lines : y = m(x-a)+b ⇔ mx - y - am + b = 0, where m represents the gradient of the the tangent lines.

the distance between the point (0,0) which is the center of the circle and the tangent lines is

|-am+b|

--------------

root(m^2+1)

= r

⇔  (a^2-r^2)m^2-2abm+b^2-r^2=0

⇔  m =

ab ± root(r^2(a^2+b^2-r^2))

-------------------------------------

a^2-r^2

therefore, the tangent lines are

y =

ab ± root(r^2(a^2+b^2-r^2))

------------------------------------- (x-a) + b

a^2-r^2

<question>

find the tangent lines through a point (-4,1) out of a circle x^2+y^2-4x-2y-4=0.

<answer>

x^2+y^2-4x-2y-4=0 ⇔ (x-2)^2+(y-1)^2=9

moving the center of the circle (2,1) to (0,0),

the point (-4,1) to(-6,0).

therefore

m =

0 ± root(9(36+0-9))

-------------------------------

36-9

= ±1/root(3)

therefore, by m=±1/root(3) and (-4,1) ,

y = ±1/root(3)(x + 4) + 1

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