I don't know how to solve for the Area and Circumference but I know how to solve for the equation but I just wanted to make sure... Any help and explanations would be appreciated :)

Problem: A circle is centered at (2, 1) and tangent to the line x + y=0. Find the general equation of the circle, the area and circumference...

  • $\begingroup$ How far is the centre to the tangent line? $\endgroup$ – peterwhy Nov 29 '14 at 10:31
  • $\begingroup$ If you have got the equation (which is probably from the fact that the perpendicular distance of the centre from the tangent is equal to the radius), then you have the radius. So, just find $2\pi r$ for circumference and $\pi r^2$ for area. $\endgroup$ – Tejas Nov 29 '14 at 13:21
  • $\begingroup$ Hint: See distance from a point to a line, then use what has already been pointed out in the other two comments. $\endgroup$ – Lucian Nov 29 '14 at 15:25

Let the radius of the circle be r. Then, its equation is $(x – 2)^2 + (y – 1)^2 = r^2$ … (1)

It cuts the line $x + y = 0$ …. (2)

Eliminate y from (1) and (2) to obtain a quadratic equation in x (with the unknown r) … (3)

(2) is tangent to (1) means (3) has only one root. Therefore, solving ⊿ = 0, we get the value of $r^2$.

Then, re-write (1).

  • $\begingroup$ the way it is supposed to be done. you could also find the slope and the equation of the radius at the point of contact. then solve the two linear equations for the contact point. from there you get the radius etc. $\endgroup$ – abel Dec 2 '14 at 5:11
  • $\begingroup$ @abel Thanks. Some have included your suggestion in their comment(s) already. I just want to add another possible way to solve the same problem. $\endgroup$ – Mick Dec 2 '14 at 14:17

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