Study the relative position of the circumference $$x^2+y^2-2x-3=0$$ and the straight line $$3x+y-5=0$$.
Development:
We have the following system of two equations: $$$\left\{\begin{array}{c} x^2+y^2-2x-3=0 \\ 3x+y-5=0 \end{array}\right\} \Rightarrow x^2+(5-3x)^2-2x-3=0 \Rightarrow 5x^2-16x+11=0$$$
We compute the discriminant of the equation of the second degree: $$$x=\dfrac{16\pm\sqrt{16^2-4\cdot5\cdot11}}{2\cdot5}=\dfrac{16\pm\sqrt{36}}{10} \Rightarrow \Delta=36 > 0$$$ The straight line and the circumference are secant since the discrimant is greater than zero.
Let's calculate the two intersection points. $$$x=\dfrac{16\pm\sqrt{36}}{10} \Rightarrow x=\left\{\begin{array}{c} \dfrac{11}{5} \\ 1 \end{array}\right. \Rightarrow y=\left\{\begin{array}{c} \dfrac{-8}{5} \\ 2 \end{array}\right.$$$
Solution:
They are secant at points of the circumference $$\Big(\dfrac{11}{5},-\dfrac{8}{5}\Big)$$ and $$(1,2)$$.