Help find eqn of circle given another circle that is tangent

In summary, to find the standard equation of a circle passing through a given point and containing the points of intersection of two circles, you first need to find the points of intersection of the two given circles. Then, you can find the circumcircle of the given point and the two intersection points. This will give you the standard equation of the desired circle.
  • #1
sktrinh
3
0
Please help me find the standard equation of the circle passing through the point (−3,1) and containing the points of intersection of the circles

x^2 + y^2 + 5x = 1

and

x^2 + y^2 + y = 7

I don't know how to begin, I am used to tangent lines or other points, but I don't know what is visually going on here. I can find the two centres C(h,k) of the given equations (-5/2,0) & (0,-1/2) both with r = sqrt(29/4), but what is the conceptual trick to equate that to the equation in question? Thanks for your help.
 
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  • #2
Re: help find eqn of circle given another circle that is tangent

I would begin by finding the points of intersection of the two given circles. We know these points will lie on the line perpendicular to the line containing the centers and that is midway between the centers.
 
  • #3
Re: help find eqn of circle given another circle that is tangent

Hi,
I agree with Mark for your specific circles. In general for two given circles and a point P not on the line of intersection of the circles, you want to find the equation of the circle passing through the intersection points and P. To do this first find the intersection points Q and R of the two circles. Then find the circumcircle of points P, Q and R. Here's an excellent web page that, among other things, gives algorithms for these two problems:
Circle, Cylinder, Sphere
 

FAQ: Help find eqn of circle given another circle that is tangent

1. How do I find the equation of a circle using another circle that is tangent?

To find the equation of a circle using another circle that is tangent, you will need to use the distance formula and the Pythagorean theorem. First, find the distance between the centers of the two circles. Then, use the Pythagorean theorem to find the radius of the new circle. Finally, use the center coordinates and radius to write the equation of the new circle.

2. What information do I need in order to find the equation of a circle using another circle that is tangent?

In order to find the equation of a circle using another circle that is tangent, you will need the center coordinates of both circles and the radius of the original circle.

3. Can I use any other methods besides the distance formula and Pythagorean theorem to find the equation of a circle using another circle that is tangent?

Yes, there are other methods such as using the tangent line equation and the slope formula. However, the distance formula and Pythagorean theorem are the most common and easiest methods.

4. What if the two circles are not tangent to each other?

If the two circles are not tangent to each other, you will not be able to find the equation of the new circle using the given information. You will need additional information, such as the coordinates of a point on the new circle, to find the equation.

5. Is there a specific order I should follow when using the distance formula and Pythagorean theorem to find the equation of a circle using another circle that is tangent?

Yes, it is important to follow a specific order when using these formulas. First, find the distance between the center points of the two circles. Then, use the Pythagorean theorem to find the radius of the new circle. Finally, use the center coordinates and radius to write the equation of the new circle.

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