# What would be the nature of tangent line when the point is inside, outside and on the circle?

Can any one tell me about the nature of line when the point is inside, outside or on the circle. I Know when a point is inside the circle, line touches the two points of circle, and when the point is on the circle then line touches the circle at one point and when the point is outside then it does not touch the circle.. Can any one clear this graphically.

Thanks.

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You call it a tangent, only when the point is 'on' the circle. When inside, we call it a secant, and when outside, well, I don't know if it has a special name. –  Shubham May 24 at 16:17

If a point is inside the circle, then there are no lines through the point that are tangent to the circle. If it is outside the circle, there are two lines through point that are tangent to the circle. If it is on the circle, there is exactly one line through the point tangent to the circle. Here's a picture:

The point A is outside the circle, and as you can see there are two lines through it, tangent to the circle. The point B is on the circle, and there is exactly one line through it tangent to the circle. (Likewise for point C.) The point D is inside the circle. There are infinitely many lines passing through it; I've drawn one. All these lines through D intersect the circle in two points, but none of them are tangent to the circle.

I assume, by the way, that by "touching" you mean "tangent to". If not, then please elaborate.

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Thanks Michael Weiss. I meant same that you have told. Thank You.. –  zonnie May 24 at 17:21

Given that C is a circle whose equation is $x^2 + y^2 + Dx + Ey + F = 0$.

Let P(h, k) be any point (that could be inside, on or outside of the circle) on the plane.

By considering $s = h^2 + k^2 + Dh + Ek + F$, one can tell whether P is inside/on/outside of the circle.

If $s > 0, P$ is outside; if $s = 0, P$ is on C; and if $x < 0, P$ is inside of the circle.

Furthermore, if $s > 0$, $\sqrt s$ gives the length of that tangent.

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