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What is the equation of circle with radius $\sqrt{2}$, tangent to the line $x+y=3$, and having its center on the line $y=4x$?

Can someone help me please?

N. F. Taussig
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Lester Rodriguez
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    The best thing to do is focus on finding the center of the circle. Once you have that, the equation of the circle will be easy to write out. Have you tried? – hardmath Mar 10 '16 at 12:47
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    Hint: let the center be of the form $(x_0,y_0)$ and use the fact that the distance between center and tangent line is the radius. You will have two equations which you should solve. Also, for the next time, it is highly recommended to share some of your thoughts. – Galc127 Mar 10 '16 at 12:49
  • Why the negative votes? I voted one up. – Friedrich Philipp Mar 10 '16 at 12:52
  • @FriedrichPhilipp I suspect that the downvotes were cast since Lester did not indicate what he attempted or where he was stuck. – N. F. Taussig Mar 10 '16 at 12:53
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    Many do that and don't get any downvotes. This is something you can also tell him by commenting. In addition, Lester seems to be new. – Friedrich Philipp Mar 10 '16 at 12:55
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    This downvoting thing isn't a very nice feature in this site, in particular when there is not even a little explanation about why. – DonAntonio Mar 10 '16 at 12:56
  • I completely agree with you. On the other hand, people post complete solutions as done below. This does not go together really well. – Friedrich Philipp Mar 10 '16 at 12:57
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    @FriedrichPhilipp I did not cast a downvote myself. It seems unreasonable to cast a downvote without explanation on a question by a new user without first explaining how he or she could improve the question to meet the site's standards. – N. F. Taussig Mar 10 '16 at 12:58
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    And now the question's been closed. I really don't get it: is it too hard to understand that sometime people's completely lost? I've voted to reopen – DonAntonio Mar 10 '16 at 13:01
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    @FriedrichPhilipp, Joanpemo etc. As I hope you're aware, the issue has been (hotly) debated many times over on [meta]. The main point to remember is that different people envision different goals for the site---and are entitled to do so. – Danu Mar 10 '16 at 13:04
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    Are you familiar with the formula $$d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}} - r$$ for the distance between the line $ax + by + c = 0$ and the circle $(x - x_0)^2 + (y - y_0)^2 = r^2$? – N. F. Taussig Mar 10 '16 at 13:05
  • @Danu Thank you. I'm aware they're entitled do...and so am I, hopefully, to my opinion. – DonAntonio Mar 10 '16 at 16:55

1 Answers1

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For the circle's center you want a point on the line $\;y=4x\;$, namely of the form $\;(a,4a)\;$ , and a distance of $\;\sqrt2\;$ from the line $\;x+y-3=0\;$ :

$$\frac{|a+4a-3|}{\sqrt{1+1}}=\sqrt2\iff |5a-3|=2\iff\begin{cases}-5a+3=2\iff a=\frac15\\{}\\5a-3=2\iff a=1\end{cases}$$

Complete now.

DonAntonio
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