find tangent line to the circle x^2+y^2+10x-6y-2=0 parallel to y=2x.please don't use calculus, i need to understand this using geometry only. hope you could help me
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We have to find the tangent to the circle x^2 + y^2 + 10x - 6y - 2 = 0 parallel to y = 2x without using calculus.
The given circle is
x^2+y^2+10x-6y-2=0
=> x^2+ 10x + 25 + y^2 - 6y + 9 = + 25 + 9 + 2
=> (x + 5) + (y - 3) = 6^2
The center of the circle is ( -5 , 3) and the radius is 6
Let the required line be y = 2x + c
The distance of (-5, 3) from 2x - y + c = 0 is 6
|2*(-5) - 3 + c| / sqrt ( 4 + 1) = 6
=> | -10 - 3 + c| = 6* sqrt 5
=> |-13 + c| = 6* sqrt 5
=> -13 + c = 6* sqrt 5
=> c = 6* sqrt 5 + 13
or 13 - c = 6* sqrt 5
=> c = 13 - 6* sqrt 5
The equations of the tangents are 2x - y + 6*sqrt 5 + 13 = 0 and 2x - y + 13 - 6*sqrt 5 = 0
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