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Question 1014205: The circle is tangent to the line 3x -4y =34 at point (10,-1) and also tangent to the line 4x+3y=12 at point (3,0). Find the equation of the circle in general form
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! 1) A tangent to a circle is perpendicular to the radius at the point of tangency
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2) The slope of the radius is the negative reciprocal of the tangent line's slope
We have two lines
3x -4y = 34 and 4x +3y = 12, solve each one for y
y = 3x/4 -17/2 and y = -4x/3 + 4
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3) now we can write two equations for the radius line
y = -4/3 x + b
y = 3x/4 + b
now use the point of tangency for each line to determine the b
-1 = -4(10)/3 + b
-3 = -40 + 3b
3b = 37
b = 37/3
y = -4x/3 + 37/3
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0 = 3*3/4 + b
b = -9/4
y = 3x/4 + 9/4
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4) set the two lines = to each other and solve for x
-4x/3 + 37/3 = 3x/4 + 9/4
-16x + 148 = 9x + 27
25x = 121
x = 121 / 25 = 4.84
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5) substitute for x in either equation for r to get the y coordinate
pick y = 3x/4 + 9/4
y = 3(4.84) / 4 + 9/4
y = 0.75*4.84 + 2.25
y = 5.88
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6) center of circle is (4.84, 5.88)
to find r use distance formula from center to point of tangency, pick (3, 0)
distance(d) = sqrt((4.84-3)^2 + 5.88^2)
d = 6.16116872 approx 6.16
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7)general form of circle is
(x-4.84)^2 + (y-5.88)^2 = 6.16^2
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