How do I find the position of the point on the circumference tagged as ?,? in terms of x and y
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Denote the unknown point $\,(a,b)\,$ , then we have the vectors $$A:=(x,y-2)-(x,y)=(0,-2)\,\,\,,\,\,\,B:=(a,b)-(x,y)=(a-x,b-y)$$ and now using the relation between inner product and $\cos\,$ we get: $$\frac{\sqrt 3}{2}=\cos 30=\frac{A\cdot B}{||A||\,\,||B||}=\frac{2(y-b)}{2\cdot \sqrt{(x-a)^2+(y-b)^2}}\Longrightarrow$$ $$\Longrightarrow (x-a)^2+(y-b)^2=\frac{4}{3}(y-b)^2\Longrightarrow (x-a)^2=\frac{1}{3}(y-b)^2\Longrightarrow$$ $$\Longrightarrow x-a=\pm\frac{1}{\sqrt 3}\,(y-b)$$ I think you can take it from here (Note the double sign as you can have your unknown point to the right or to the left of the bottom one) |
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