Question 1191482: find the equation of the circle concentric with the circle x^2+y^2-4x+6y-17=0 which has a tangent of 3x-4y+7=0.
Found 2 solutions by ikleyn, Alan3354:
Answer by ikleyn(52847)   (Show Source):
You can put this solution on YOUR website! .
Find the equation of the concentric circle to the circle x^2+y^2-4x+6y-17=0 which has a tangent of 3x-4y+7=0.
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By completing the square method, find the center of the circle. It is the point (2,-3).
To find the radius of the required circle, find the distance from the center (2,-3) to the given line 3x-4y+7=0.
For finding the distance from a given point to a straight line, there is a remarkable formula.
Let the straight line in a coordinate plane is defined in terms of its linear equation
a*x + b*y + c = 0,
where "a", "b" and "c" are real numbers, and let P = ( , ) be the point in the coordinate plane.
Then the distance from the point P to the straight line is equal to
d =  .
Regarding this formula, see the lesson
The distance from a point to a straight line in a coordinate plane
in this site.
Your straight line is 3x - 4y + 7 = 0.
Substitute the given data a= 3, b= -4, c= 7, = 2, = -3 into the formula to get the distance under the question
 =  =  = 5.
Thus the radius of the circle should be 5 units.
Then the standard form equation of the circle is
 +  = 25. ANSWER
Solved.
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! find the equation of the circle concentric with the circle x^2+y^2-4x+6y-17=0 which has a tangent of 3x-4y+7=0.
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Find the center of the circles:
x^2+y^2-4x+6y-17=0
x^2+y^2-4x+6y = 17
(x^2 - 4x + 4) + (y^2 + 6y + 9 = 17 + 4 + 9 = 30
(x-2)^2 + (y+3)^2 = 30
Center at (2,-3)
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Find the distance from the center to the line:
The slope of the line is 3/4
The slope of lines perpendicular is -4/3
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The line thru (2,-3) with a slope of -4/3 ---> y+3 = (-4/3)*(x-2)
y + 3 = -4x/3 + 8/3
3y = -4x - 1
4x + 3y = -1
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Find the intersection with 3x-4y+7=0
3x-4y = -7
4x+3y = -1
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9x - 12y = -21
16x +12y = -4
---------------------- Add
25x = -25
x = -1
-3 - 4y = -7
-4y = -4
y = 1
---> the intersection is (-1,1)
Find the distance from the center at (2,-3) to (-1,1)
d^2 = diffy^2 + diffx^2 = 4^2 + 3 = 25
---> the circle is (x-2)^2 + (y+3)^2 = 25
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