Question 1133939
Substitute {{{y=3x+k}}} into {{{x^2+y^2-8=0}}}

{{{x^2+(3x+k)^2-8=0}}}
{{{x^2+9x^2+6kx+k^2-8=0}}}
{{{10x^2+6kx-8=0}}}

Using the formula {{{b^2-4ac<0}}} as it doesn’t intersect the curve.
{{{(6k)^2-4(10)(k^2-8)<0}}}
{{{36k^2-40k^2+320<0}}}
{{{-4k^2+320<0}}}
{{{4k^2>320}}}

Separate into possible cases,
{{{k>sqrt(80)}}} and {{{-k>sqrt(80)}}}

Hence, the range of values of k,
{{{k>sqrt(80)}}} , {{{k<-sqrt(80)}}}