SOLUTION: What is the equation of the circle with center at(0.2) and tangent to the line 3x-4y=12

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Question 1187084: What is the equation of the circle with center at(0.2) and tangent to the line
3x-4y=12

Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
What is the equation of the circle with center at(0.2) and tangent to the line
3x-4y=12
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There's a formula for the distance from a point to a line, but I'll work it the "long way."
---------
3x-4y=12 ---> y = (3/4)x - 3
Slope of the line is 3/4
Slope of lines perpendicular is -4/3
(0,2) is the y intercept ---> y = (-4/3)x + 2
Find the intersection of the 2 lines.
---
y = (3/4)x - 3
y = (-4/3)x + 2
---
(3/4)x - 3 = (-4/3)x + 2
9x - 36 = -16x + 24
25x = 60
x = 2.4
y = -1.2
Intersection at (2.4,-1.2)
Distance from (0,2) = sqrt(diffy^2 + diffx^2) = sqrt(2.4^2 + 3.2^2) = sqrt(16) = 4
-------
is the circle

Answer by ikleyn(52845)   (Show Source): You can put this solution on YOUR website!
.
What is the equation of the circle with center at (0,2) and tangent to the line 3x-4y=12.
~~~~~~~~~~~~~~~~~~ 


All you need to do is to find the distance from the point (0,2) to the given straight line  3x - 4y - 12 = 0.


There is a remarkable formula which ideally suits for this need.


    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.


Substitute the given data  a= 3, b= -4, c= -12,   = 0,  = 2  into the formula to get the distance under the question


     =  = 4.


Thus the radius of the circle is 4 units.


THEREFORE, the circle equation is


     +  = ,

or

     +  = 16.      ANSWER

Solved.



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