I'm reading a book about Calculus on my own and am stuck at a problem, the problem is

There are two circles of radius $2$ that have centers on the line $x = 1$ and pass through the origin. Find their equations.

The equation for circle is $(x-h)^2 + (y-k)^2 = r^2$

Any hints will be really appreciated.

EDIT: Here is what I did. I drew a triangle from the origin and applied the pythogoras theorem to find the perpendicular, the hypotenuse being $2$ ($\text{radius} = 2$) and base $1$ (because $x = 1$), the value of y-coordinate is $\sqrt3$. Can anyone confirm if this is correct?

  • $\begingroup$ For your edit: that gives only one circle. How about the other one? $\endgroup$ – J. M. is a poor mathematician Dec 27 '10 at 3:43

Hint: The distance between a point $P(h,k)=P(1,k)$ and the origin $O(0,0)$ is given by $\sqrt{1+k^{2}}$. This distance should be equal to $2$.

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    $\begingroup$ what did you use to make the diagram? $\endgroup$ – Arjang Dec 27 '10 at 3:47
  • $\begingroup$ Thanks. The answer is sqrt(3). I got the same answer by applying the Pythagoras theorem. $\endgroup$ – Cixet Dec 27 '10 at 4:00
  • $\begingroup$ Same question: which software do you for quick drawing? $\endgroup$ – user2468 Dec 27 '10 at 5:09
  • $\begingroup$ @Arjand, @M.S., I used Scientific WorkPlace from mackichan.com $\endgroup$ – Américo Tavares Dec 27 '10 at 12:27

HINT: Find out what points $(x, y)$ with $x = 1$ also have distance $2$ to the origin. What would these points represent?

  • $\begingroup$ what is (x|y) with x = 1 ? $\endgroup$ – Cixet Dec 26 '10 at 17:29
  • $\begingroup$ @Cixet: it’s the predominant German notation for two-dimensional points $(x, y)$. $\endgroup$ – Konrad Rudolph Dec 26 '10 at 19:15
  • $\begingroup$ @Konrad: Oh thanks. By the way, is $\vec{x}$ also specifically German? $\endgroup$ – Dario Dec 26 '10 at 19:50
  • $\begingroup$ nope. [ ](example.com) $\endgroup$ – Konrad Rudolph Dec 26 '10 at 20:31

Note that you can write down the equation of a circle if you know the co-ordinates of the center $(h,k)$ and its radius $r$. You already know that the two circles in question have radius $2$. It remains to figure out where their centers lie. You are told that their centers lie on the line $x=1$ which is a line parallel to the $Y$-axis. So you know the $x$-coordinate of the centers. The only thing that remains to be figured out are the $y$-coordinates of the two circles. To figure this out, you are given an additional information that both circles pass through the origin.

I would suggest drawing a picture of the Cartesian plane and of the line $x=1$ on it. You know the centers lie on this line and you know the centers have to be at a certain distance from the origin (why?). Given these two constraints, figure out what possible locations the centers can be at. If this isn't clear enough, post your work and indicate where you are getting stuck.

  • $\begingroup$ To be honest, I haven't done any work because I have no idea what to do, how can I work out the possible locations for y-coordinate? $\endgroup$ – Cixet Dec 26 '10 at 17:28
  • $\begingroup$ @Cixet: Let the centers be of the form $(h,k)$. Then you can figure out the value of $h$ immediately. To find $k$, use the fact that you know the distance of the point $(h,k)$ from the origin. Again, drawing a picture would help a lot in understanding this. $\endgroup$ – Dinesh Dec 26 '10 at 17:57

Rephrasing Dario's hint: the center of each circle is a point of distance $2$ from the origin with $x=1$. If the center is the point $(x,y)$ then $x=1$, and so you can solve for $y$ (remember, the distance between $(x,y)$ and $(0,0)$ is $2$).


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