What I want to know is if I have a unit circle centered at the origin with a diameter drawn along the y axis and I know the distance that a point on the diameter has from the center what would be the x coordinates of the points on the circumference with the same y coordinates. further more how would you repeat this process on bigger or smaller circles.
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I didn't quite understand your question. You want to find the coordinates of points that have the same y-coordinate and lie on the circumference of the unit circle? – Jamāl Dec 27 '20 at 14:36
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Yes that is my question. – Award Mathpu Dec 27 '20 at 14:39
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Say the circle with radius $r$ has equation $$x^2+y^2 =r^2 $$ and the point you have is $P(0,a)$ ($a$ is the distance of the point from the center) where $-r\le a\le r$. Then you want to solve for the intersection point of $y=a$ with the circle: $$x^2+a^2 =r^2 \implies x=\pm\sqrt{r^2-a^2}$$
Vishu
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also i think the OP is asking for the points that have the same y-coordinate as the point $P$ on the diameter along the y-axis. – Jamāl Dec 27 '20 at 14:55
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Alternatively, you can solve this using trigonometry.
Let's say the point along the diameter (diameter being twice the radius, $r$) above the origin is $y$, $(0,y)$. The angle the swept radius makes from the horizontal axis so that the $y$ component of the triangle is,$$y=r\cdot \sin(\theta)$$The points of this triangle being: $(0,0), (x,0),$ and $(x,y)$.
We can solve for $\theta$ using $$\theta = \sin^{-1}\left(\frac{y}{r}\right)$$ Then, we can solve for the x-component using$$x=r\cdot \cos(\theta)$$ or,$$x=\frac{y}{\tan(\theta)}$$ Since you want both sides of the x-axis, the xs you are looking for are $\pm x$ of the above.
Jim Clark
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