SOLUTION: find equation of the circle that satisfies the given condition A. tangent 3x-4y=10 and center at c(0,5)

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Question 1085055: find equation of the circle that satisfies the given condition
A. tangent 3x-4y=10 and center at c(0,5)
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Find the perpendicular line that contains the center.
4y=3x-10
y=%283%2F4%29x-5%2F2
Perpendicular lines have slopes that are negative reciprocals,
%283%2F4%29m%5B2%5D=-1
m%5B2%5D=-4%2F3
Use the point slope form of a line,
y-5=-%284%2F3%29%28x-0%29
y=-%284%2F3%29x%2B5
Find the intersection point of the two lines,
+%283%2F4%29x-5%2F2=-%284%2F3%29x%2B5+
9x-30=-16x%2B60
25x=90
x=18%2F5
So then,
3%2818%2F5%29-4y=10
54-20y=50
-20y=-4
y=1%2F5
Use the distance formula to find the distance between the two points (the radius of the circle),
R%5E2=%2818%2F5-0%29%5E2%2B%281%2F5-5%29%5E2
R%5E2=36%2F25%2B%28-24%2F5%29%5E2
R%5E2=324%2F25%2B576%2F25
R%5E2=900%2F25
R%5E2=36
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%28x-0%29%5E2%2B%28y-5%29%5E2=4
highlight%28x%5E2%2B%28y-5%29%5E2=36%29
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Answer by ikleyn(52914) About Me  (Show Source):
You can put this solution on YOUR website!
.
find equation of the circle that satisfies the given condition
A. tangent 3x-4y=10 and center at c(0,5)
~~~~~~~~~~~~~~~~~~ 

1.  Find the distance from the point (0,5) to the given line 3x - 4y = 10.


    For it, use the formula on the distance from the given point to the given line of the lesson
        The distance from a point to a straight line in a coordinate plane 
    in this site.


    You will get

    distance = abs%283%2A0%2B%28-4%29%2A5%29%2Fsqrt%283%5E2%2B4%5E2%29 = 20%2F5 = 4.


    Thus you found the radius of the circle. It is 4 units.


2.  Then the equation of the circle is 

    %28x-0%29%5E2+%2B+%28y-5%29%5E2 = 4%5E2,     or


    x%5E2+%2B+%28y-5%29%5E2 = 16.


The solution by the other tutor is not correct, unfortunately.


To see more solved problem of this type, look into the lesson
    - Find the standard equation of a circle
in this site.


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic
"Conic sections: Ellipses. Definition, major elements and properties. Solved problems".