Find the equation of the circle which touches the $X$ axis at the point $(4,0)$ and passes through the point $(2,2)$ .

My Attempt:

since the circle touches the $x$ axis, y- co ordinate of the centre is equal to the radius of circle. But, I don't see the centre of the circle given here. How could this be solved?

  • $\begingroup$ What's the general equation of a circle ? $\endgroup$ – user399481 Dec 27 '16 at 7:23
  • 1
    $\begingroup$ ##Fib1123, Its $x^2+y^2+2gx+2fy+c=0$. $\endgroup$ – pi-π Dec 27 '16 at 7:26
  • 2
    $\begingroup$ You may alternatively write $(x-a)^2+(y-b)^2=r^2$ $\endgroup$ – Mythomorphic Dec 27 '16 at 7:28
  • $\begingroup$ First, edit your question. Some words missing. Then use the given information to determine the unknowns. $\endgroup$ – user399481 Dec 27 '16 at 7:29
  • $\begingroup$ @user354073 also use the fact that the $x$ coordinate of the center is equal to the $x$ coordinate of the point where it couches the X axis $\endgroup$ – Andrei Dec 27 '16 at 7:29

It's easiest to work with the equation of the circle given by

$$(x-a)^2 + (y-b)^2=r^2$$

as this is the circle with center $(a,b)$ and radius $r$.

You are given two facts:

  1. The circle touches the $x$ axis at $(4,0)$.
  2. The circle passes through the point $(2,2)$.

The first fact tells you that the center of the circle must have an $x$ coordinate of $4$ and that the radius of the circle is the same as the $y$-coordinate.

Your job: Try to put the two things into equations. Knowing that $a$ is the $x$-coordinate and $b$ the $y$-coordinate of the center, and that $r$ is the radius, what equation does the sentence "The $x$ coordinate of the center is $4$" describe? What about the sentence "The radius is the same as the $y$ coordinate"?

The second fact tells you that plugging in $x=2$ and $y=2$ into the circle equation will result in a true statement. This should give you the third equation (the first fact gives you two), and you will have $3$ equations for $3$ variables, $a,b$ and $r$.


$(x-4)^2+(y-2)^2=2^2$. Just graph it and convince yourself why this should be one of the answer satisfying your requirements.

  • $\begingroup$ How is the centre $(4,2)$? $\endgroup$ – pi-π Dec 27 '16 at 7:34
  • $\begingroup$ I have given for the case when X axis is the tangent. $\endgroup$ – user1131274 Dec 27 '16 at 7:39
  • $\begingroup$ I did not see that $x$-axis is the tangent line at $(4,0)$. $\endgroup$ – Juniven Dec 27 '16 at 7:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.