Question 1184256
.



            In her post,  @MathLover1  INCORRECTLY  interprets the problem,

            considering only the case when the straight line touches the circle.


            The correct interpretation is to consider a general case of an intersection 
            ("having at least one intersection point").


            So,  I  came to bring a  CORRECT  SOLUTION.



<pre>
{{{x^2+y^2=10x}}}....eq.1

{{{y=kx-10 }}}.....eq.2
--------------------------------

substitute {{{y}}} in eq.1

{{{x^2+(kx-10)^2 = 10x}}}

{{{x^2+k^2x^2 - 20kx -10x+ 100=0}}}

{{{(1+k^2)x^2 - (20k +10)x+ 100=0}}}


if the line meets the curve, there is <U>at least</U> one solution and it is a case if discriminant >= {{{0}}}


so, use discriminant 

{{{b^2-4ac >= 0}}}  ...in your case {{{a=(1+k^2)}}}, {{{b=- (20k +10)}}}, and {{{c=100}}}

{{{(- (20k +10))^2-4(1+k^2)*100 >= 0  }}}

{{{400k^2 + 400k + 100-400 k^2 - 400 >= 0 }}}
 
 {{{400k - 300 >= 0  }}}

{{{ 400k >= 300 }}}
 
{{{4k >= 3}}}

{{{k >= 3/4}}}


<U>ANSWER</U>.  k >= {{{3/4}}}.
</pre>

Solved (correctly).