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If a circle touches a point (like any point on the co-ordinate axes), does it mean that the point will satisfy the equation of the circle?

Also, if the equation of the line on which the centre of the circle is given, then should the perpendicular distance of the point from the line, be the radius of the circle?

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Consider the above circle with centre A, on the circumference of which there lies a point, B. Now, this point can lie on any arbitrary straight line, which will then become a tangent to the circle, and the point B would become the point of tangency.

Now, the radius dropped on a tangent is always perpendicular to the tangent, and the vice-versa is also correct, that is, the radius joining the point of tangency to the line on which the centre A lies, is a normal to the line. So, the line BA, which is perpendicular to the line on which the centre A lies (ignore inaccuracies in diagram), is a radius, right?

Wrichik Basu
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  • Yes to the first. Yes also to the second, assuming I understand it. I think you are asking "suppose a line $L$ is tangent to a circle. If I drop a perpendicular from the center of the circle to $L$ is it a radius?" If I have that right, then the answer is yes. – lulu Sep 09 '17 at 16:47
  • @lulu edited to clarify. Please see. – Wrichik Basu Sep 10 '17 at 05:52
  • I don't understand what you wrote. By definition, the segment $AB$ is a radius as it connects the center with a point on the circle. It does not follow that $AB$ is perpendicular to $AD$. (For instance, $AD$ is not perpendicular to $AD$). – lulu Sep 10 '17 at 09:54

1 Answers1

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Question 1: Yes. That is what the equation of a circle means, after all. It describes the specific relation between the $x$ and $y$ coordinates of points in the circle. Whether that point lies on some other line is irrelevant.

Question 2: No. The point may even be on the line. However, the perpendicular distance from the point to the line can never be larger than the radius.

Edited question 2: Yes, any line (segment) from the center of the circle to the periphery is a radius, and the perpendicular distance from the center to a tangent line is equal to the radius.

Arthur
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  • Couldn't understand the second part. The radius to a tangent is a normal to the point of tangency. I'd I consider the point on the circumference to be a point of tangency, (because any point can lie on a line, which will then become a tangent to the circle), then won't the line joining the point and the radius be a normal to a line on which the radius lies? – Wrichik Basu Sep 09 '17 at 17:55
  • In that case I misunderstood what you meant with "the line on which the centre of the circle". I thought you meant "Given a line on which the centre of the circle lies, is the distance between that line and a point on the circle equal to the radius?" If that's not it, you will have to be a bit clearer, and I will try to answer. – Arthur Sep 09 '17 at 17:59
  • Question edited. Have a look. – Wrichik Basu Sep 09 '17 at 18:00