A circle with radius $5$ and center $(a,b)$ is tangent to the lines $y = 6$ and $y = x.$ Compute the largest possible value of $a + b.$
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y = 6
y = x
P(6,6)
\(\color{blue}m_{f(x)}=tan22.5^{\circ}\\f(x)=m(x-x_P)+y_P\\ f(x)=tan\ 22.5^{\circ}(x-6)+6\\ f(x)=tan\ 22.5^{\circ}\cdot x-6\cdot tan\ 22.5^{\circ}+6\\\color{blue} f(x)=tan\ 22.5^{\circ}\cdot x+6(1- tan\ 22.5^{\circ})\)
\(\color{blue}g(x)=6-5=1\\\color{blue} g(x)=f(x)\\ 1=tan\ 22.5^{\circ}\cdot x+6(1- tan\ 22.5^{\circ})\\ x=\dfrac{1-6(1- tan\ 22.5^{\circ})}{ tan\ 22.5^{\circ}}\\ x=\color{blue}a=-6.071\\ y=\color{blue}b=1\\ \color{blue}a+b=-5.071\)
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