[tex]\it AB^2 =(x_B-x_A)^2+(y_B-y_A)^2=12^2+a^2\ \ \ \ \ \ (1)\\ \\ AB=13 \Rightarrow AB^2=13^2\ \ \ \ \ \ (2)\\ \\ (1),\ (2) \Rightarrow 12^2+a^2=13^2 \Rightarrow a^2=13^2-12^2=(13-12)(13+12)=25 \Rightarrow \\ \\ \sqrt{a^2}=\sqrt{25} \Rightarrow |a|=5 \Rightarrow a=\pm5[/tex]
