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Given that radius of $C_2$ is $\sqrt 2$, equation of $C_1$ is $x^2 - 4x + y^2 -8y=0$ and $C_2$ intersects $C_1$ at origin and where $x=2$, find the coordinates of centre of $C_2$ if the centre of $C_2$ is in $C_1$.

I've found that the second point of intersection is $(2, 4-2\sqrt 5)$, tried letting the coordinates of the centre be $(a, b)$, then coming up with 2 equations whereby the euclidean distance from the centre of the $C_2$ and the two points of intersection equals $\sqrt 2$.

However I end up with 2 equations with unknown variables $a, b$ to the power of 2 and am unsure of how to solve from here on.

Any help will be greatly appreciated thank you.

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The center of those circles that cross the origin and whose radius is $\sqrt 2$ sit on a circle of radius $\sqrt 2$ centered at the origin. So our circles are described by$$(x-a)^2+(y-b)^2=2.$$ For their centers we have $$a^2+b^2=2\ \ \Rightarrow b=\sqrt{2-a^2}$$ so, finally, the equations of our circles are $$(x-a)^2+(y-\sqrt{2-a^2})^2=2.$$ I consciously omitted the minus sign above because we are interested only in those circles whose centers are located in the first quadrant. The following figure depicts the two extremes:

enter image description here

Intuition tells that there exists a little circle crossing the big one at a point whose $x=2$. The center of the required circle is centered on the goose poop colored circle somewhere between the two black dots (in the first quadrant.)


The $y$ coordinate of the point on the large circle is $$4-\sqrt{20}$$ if $x=2$. I consciously omitted the $+$ sign because our $y$ is negative.

Now, the question is: which point of the goose poop colored central circle is $\sqrt 2$ units away from the point $(2,4-\sqrt{20})$?

In the first quadrant the coordinates of the points on the central circle are $(a,\sqrt{2-a^2})$. The square of the distance of such a point from $(2,4-\sqrt{20})$ is

$(a-2)^2+(\sqrt{2-a^2}-(4-\sqrt{20}))^2$ then the equation one will have to solve is

$$(a-2)^2+(\sqrt{2-a^2}-(4-\sqrt{20}))^2=2.$$

Alpha solved the equation above for me and the solution turned out to be equal to the solution given by the OP in a comment to Michael's answer.

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If "the center of $C_2$ is in $C_1$" it means "the centre of $C_2$ is placed on $C_1$" then we have the following hint.

For the $C_2(a,b)$ we have the following system. $$a^2+b^2=2$$ and $$a^2+b^2-4a-8b=0.$$

Now, check that the distance between $(a,b)$ and $(2,4\pm2\sqrt5)$ is equal to $\sqrt2$, which gives that $C_2$ does not exist.

If it means "the center of $C_2$ is inside of $C_1$" then the answer indeed is very ugly.

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  • $\begingroup$ I believe it means the latter- "center of $C_2$ is inside of $C_1$". I have the answer key- it is supposed to be $(1.22, 0.710)$, but I can't get another equation to solve for both $a$ and $b$. $\endgroup$ – user445310 Dec 16 '17 at 8:49
  • $\begingroup$ @Michael Rozenberg: Yes, the circle in question exists. I calculated the coordinates of its center. $\endgroup$ – zoli Dec 16 '17 at 9:35
  • $\begingroup$ I am not ready for this ugly work. $\endgroup$ – Michael Rozenberg Dec 16 '17 at 9:47

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