MHB Tangent line to circle making 30º with a second circle

AI Thread Summary
The discussion focuses on finding a tangent line to circle B that intersects circle A at a 30º angle. Circle A is located at coordinates (479183.87, 4365099.87) with a radius of 27780m, while circle B is at (488889.66, 4390316.69) with a radius of 5294m. A detailed mathematical approach is provided, involving trigonometric calculations to determine the angles and coordinates of the intersection points. The final results yield approximate coordinates for the points of intersection on both circles. The explanation is well-received, confirming it meets the original query's requirements.
GONURVIA
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Hi all, this is my first thread!

I am having problems trying to find the way of drawing a line which is tangent to a circle and intersects another circle making a 30º intersection.

Let´s say I have circle A with coordinates 479183.87, 4365099.87 (x1,y1) and a radius of 27780m. I have a second circle, B, with coordinates 488889.66, 4390316.69 (x2,y2) and a radius of 5294m. I would like to draw a line which joins both circle, being the line tangent to circle B and intersects circle A with 30º to its circumference. I've attached an image hoping it helps visually to figure out what I'm trying to ask for

I hope I explained myself correctly. Thanks everyone!
 

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GONURVIA said:
Hi all, this is my first thread!

I am having problems trying to find the way of drawing a line which is tangent to a circle and intersects another circle making a 30º intersection.

Let´s say I have circle A with coordinates 479183.87, 4365099.87 (x1,y1) and a radius of 27780m. I have a second circle, B, with coordinates 488889.66, 4390316.69 (x2,y2) and a radius of 5294m. I would like to draw a line which joins both circle, being the line tangent to circle B and intersects circle A with 30º to its circumference. I've attached an image hoping it helps visually to figure out what I'm trying to ask for

I hope I explained myself correctly. Thanks everyone!
Hi GONURVIA, and welcome to MHB!

[TIKZ][scale=6]
\clip (7,8.25) rectangle (10,9.6) ;
\draw (7.92,6.51) circle (2.78cm) ;
\draw (8.89,9.03) circle (0.53cm) ;
\coordinate [label=above:$C$] (C) at (8.16,9.28) ;
\coordinate [label=above:$D$] (D) at (8.666,9.512) ;
\draw (9.156,9.193) -- (C) -- (D) ;
\draw (7.92,6.51) -- (C) ;
[/TIKZ]
I assume that $(x_1,y_1)$ and $(x_2,y_2)$ are the centres of the two circles.

Let $C$ be the point where the tangent line meets circle $A$. Suppose that the radius there (the line from $(x_1,y_1)$ to $C$) makes an angle $\theta$ with the horizontal axis. Denoting the radius of circle $A$ by $R$, the coordinates of $C$ are $(x_1+R\cos\theta,y_1+R\sin\theta)$. The tangent to circle $B$ then has to be at an angle $\theta-60^\circ$ to the horizontal. So the equation of the tangent line is $$y-y_1-R\sin\theta = \tan(\theta-60^\circ)(x-x_1-R\cos\theta).$$ Using a bit of trigonometry, that becomes $$(x-x_1)\sin(\theta-60^\circ) - (y-y_1)\cos(\theta-60^\circ) + R\sin(60^\circ) = 0.$$ The condition for that line to be a tangent to circle $B$ is that the distance from $(x_2,y_2)$ to the line should be equal to $r$, the radius of circle $B$. The formula for the distance from a point to a line then gives the equation $$(x_2-x_1)\sin(\theta-60^\circ) - (y_2-y_1)\cos(\theta-60^\circ) + R\sin(60^\circ) = r.$$ (Strictly speaking, the left side of that equation should have absolute value signs round it. But since it turns out that it is already positive, I have left them out.)

Let $\alpha$ be the angle such that $\tan\alpha = \dfrac{y_2-y_1}{x_2-x_1}.$ Then a further bit of trigonometry shows that $$\sin(\theta-\alpha - 60^\circ) = \frac{r-R\sin(60^\circ)}{\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}.$$ You can then start putting in the numerical values for $x_1,x_2,y_1,y_2,R$ and $r$, to find first $\alpha$, then $\theta$ and then the coordinates for $C$.

I only have a small pocket calculator, so I can only work to a few significant figures. The results I get are $$\alpha \approx 68.95^\circ,$$ $$\theta \approx 84.97^\circ,$$ $$C \approx (481600,4392800).$$ Finally, the point $D$ where the tangent line touches circle $B$ is given by $D = (x_2 + r\cos(\theta+30^\circ), y_2+r\sin(\theta+30^\circ))$, or approximately $(486600,4395100)$.
 
Opalg, thank you very much! That was exactly what I was asking for. I appreciate the detailed explanation of how to solve the problem step by step, very helpful!
 
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