This graph shows two circles $h$ units vertically either side of the origin with common radius $r$. What is the equation of the line that passes through the origin, and is a tangent to both circles, in terms of $r$ and $h$. Approximate picture:
What is the equation of the line tangent to two equally spaced circles on the y-axis
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$\begingroup$
circles
tangent-line
4 Answers
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Hint:
The origin, the center of one of the circles and its point of tangency are the vertices of a right triangle.
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This particular case of tangency is known as tranverse common tangency. The approach to solve these kind of problems is simple Pythagorean geometry .
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The equation of the required tangent will be the line joining point $P(r,r+h)$ to the origin i.e. $y=\frac{r+h}{r}x$.
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Ok so with ajotatxe's advice I was able to see that if the slope of the line was $tan(\theta)$ then the height $h$ can be expressed as $tan(\theta)*rcos(\pi/2-\theta)+rsin(\pi/2-\theta)$ see here. Thank you everyone.