The form of the equation of the circle is
[tex](x-h)^2+(y-k)^2=r^2[/tex]
(h, k) is the center
r is the radius
Since the given center is (0, 5) and the given radius is 4, then
The equation of the circle is
[tex]\begin{gathered} (x-0)^2+(y-5)^2=16 \\ x^2+(y-5)^2=16\rightarrow(1) \end{gathered}[/tex]
Since the equation of the line is
[tex]y=1.5x+5\rightarrow(2)[/tex]
Substitute y in equation (1) by equation (2)
[tex]\begin{gathered} x^2+(1.5x+5-5)^2=16 \\ x^2+(1.5x)^2=16 \\ x^2+2.25x^2=16 \end{gathered}[/tex]
Add the like terms on the left side
[tex]3.25x^2=16[/tex]
Divide both sides by 3.25
[tex]\begin{gathered} \frac{3.25x^2}{3.25}=\frac{16}{3.25} \\ x^2=\frac{64}{13} \end{gathered}[/tex]
Take a square root for both sides
[tex]\begin{gathered} \sqrt{x^2}=\sqrt{\frac{64}{13}} \\ x=2.218800785 \end{gathered}[/tex]
Substitute the value of x in equation (2) to find y
[tex]\begin{gathered} y=1.5(2.218800785)+5 \\ y=8.328201177 \end{gathered}[/tex]
Round them to 3 decimal places
x = 2.219
y = 8.328
