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Let k be a positive real number. The line x + y = k and the circle x^2 + y^2 = 2k + 1 are drawn. Find k so that the line is tangent to the circle.

 Aug 27, 2021
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Let k be a positive real number. The line x + y = k and the circle x^2 + y^2 = 2k + 1 are drawn. Find k so that the line is tangent to the circle.

 

Hello Guest!

 

\(x^2 + y^2 = 2k + 1\\ y=\sqrt{2k+1-x^2}\\ r^2=2k+1\\ r=\sqrt{2k+1}\)

 

\(x+y=k\\ y=k-x\)

Distance of the straight line \(y=k-x\)  from the origin of the coordinates.

\(r=\frac{k}{\sqrt{2}}\)

\(\dfrac{k}{\sqrt{2}}=\sqrt{2k+1}\\ \dfrac{k^2}{2}=2k+1\\k^2-4k-2=0\\ k=2\pm \sqrt{4+2}\\ k=2\pm 2.449\)

 

\(k=4.449\)

laugh  !

 Aug 27, 2021
edited by asinus  Aug 27, 2021

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