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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?

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  • $\begingroup$ What's the general equation of a circle ? $\endgroup$ – user399481 Dec 27 '16 at 7:23
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    $\begingroup$ ##Fib1123, Its $x^2+y^2+2gx+2fy+c=0$. $\endgroup$ – pi-π Dec 27 '16 at 7:26
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    $\begingroup$ 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
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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 $r$.

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

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  • $\begingroup$ 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

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