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# plz help

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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.

$$r=\frac{k}{\sqrt{2}}$$

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

$$k=4.449$$

!

Aug 27, 2021
edited by asinus  Aug 27, 2021