SOLUTION: 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.

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