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.
+14913 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.
\(r=\frac{k}{\sqrt{2}}\)
\(\dfrac{k}{\sqrt{2}}=\sqrt{2k+1}\\ \dfrac{k^2}{2}=2k+1\\k^2-4k-2=0\\ k=2\pm \sqrt{4+2}\\ k=2\pm 2.449\)
\(k=4.449\)
!
