MHB Finding the Equation of a Circle Given Specific Conditions

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To find the equation of the circle that passes through the point (-4, 1), first determine the centers of the two given circles by rewriting their equations in standard form through completing the square. The centers can then be used to calculate the midpoint, which will serve as the center of the new circle. After finding the center, use the distance formula to determine the radius, as the circle must pass through the specified point. Finally, write the equation of the circle in standard form using the identified center and radius.
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Determine the equation of the circle that satisfies the given conditions. Write the equation in standard form.

The circle passes through (-4, 1) and its center is the midpoint of the line segment joining the centers of the two circles x^2 + y^2 - 6x - 4y + 12 = 0 and x^2 + y^2 - 14x + 47 = 0.

I would like the steps for me to solve this interesting problem.
 
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I would begin by determining the centers of the two given circles, so that we can determine the mid-point of the line segment joining those centers. So, rewrite the two given circles in the form:

$$(x-h)^2+(y-k)^2=r^2$$

What do you get?
 
MarkFL said:
I would begin by determining the centers of the two given circles, so that we can determine the mid-point of the line segment joining those centers. So, rewrite the two given circles in the form:

$$(x-h)^2+(y-k)^2=r^2$$

What do you get?

Must I complete the square for both circles?
 
RTCNTC said:
Must I complete the square for both circles?

Yes, for both circles you want to complete the square for both x and y to get them in the desired form from which you can read off the coordinates of the centers. :D
 
MarkFL said:
Yes, for both circles you want to complete the square for both x and y to get them in the desired form from which you can read off the coordinates of the centers. :D

Good. I will try later.
 
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