<s>[INST] Circle and a line that passes through it Given a line with equation: $y=ax-3$ that passes through a circle with equation $(x-1)^2+(y-1)^2= 1$. Find the range of values of $a$. 
I tried graphing and got: $0<x<2$ and $0<y<2$.
I also tried finding $a$ by substituting $x$ and $y$ into $y=ax-3$
which really confuses me.
Could you help me in solving this problem?
 [/INST] we substitute 
$y=ax-3$
in
$(x-1)^2+(y-1)^2= 1$
$(x-1)^2+(ax-3-1)^2= 1$
$(x-1)^2+(ax-4)^2= 1$
$x^2-2x+1+a^2x^2-8ax+16=1$
we have this second degree equation
$x^2(1+a^2)+x(-2-8a)+16=0$
with
$\Delta =(-2-8a)^2-4\cdot(1+a^2)\cdot 16$
if $\Delta <0$ 
the equation have no solution and the line not intersect the circle
if $\Delta =0$ 
the equation have one solution and the line is tangent to the circle
if $\Delta >0$ 
the equation have two solutions and the line intersect the circle in 2 points
$\Delta =4+32a+64a^2-64-64a^2$
$\Delta =32a-60$
so if
$32a-60>0$
$a>\frac{15}{8}$
the line intersect the circle in 2 points
and if
$32a-60=0$
$a=\frac{15}{8}$
the line is tangent to the circle 
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