Question 1200535
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Let's solve for y in the first equation
x - 2y + 4 = 0
-2y = -x-4
y = (-x-4)/(-2)
y = (1/2)x + 2
This equation has a slope of 1/2
The negative reciprocal of 1/2 is -2/1 aka -2
Therefore, the equation perpendicular to x - 2y + 4 = 0 will have a slope of -2.
Why are we looking for a perpendicular line? Because the tangent line is perpendicular to the radius when meeting at the point of tangency. Check out the diagram below. 


We'll use this perpendicular slope and the coordinates (x,y) = (0,2) to determine the equation of the perpendicular line is y = -2x+2


Another approach you can take is outlined in this lesson
<a href = "https://www.algebra.com/algebra/homework/Linear-equations/perpendicular-line-example1.lesson">https://www.algebra.com/algebra/homework/Linear-equations/perpendicular-line-example1.lesson</a> 


Through similar steps, the perpendicular line to y = 2x-7 that passes through (3,-1) is y = (-1/2)x + 1/2


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The two perpendicular lines we found
y = -2x+2
y = (-1/2)x + 1/2
will have the circle diameters located on them.


As such, those lines intersect to pinpoint the center of the circle.


y = (-1/2)x + 1/2
-2x+2 = (-1/2)x + 1/2
2*(-2x+2) = 2*((-1/2)x + 1/2)
-4x+4 = -x+1
-4x+x = 1-4
-3x = -3
x = -3/(-3)
x = 1
Then,
y = -2x+2
y = -2*1+2
y = 0
The center is located at (x,y) = (1,0)
So we know that (h,k) = (1,0)
i.e. we have h = 1 and k = 0


Those h and k values are useful for the circle template
(x-h)^2 + (y-k)^2 = r^2


Diagram
*[illustration Screenshot_80.png]
I used GeoGebra, but Desmos is another good option.


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Answer: Center is <font color=red>(1,0)</font>
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