$$\begin{aligned} &a(b-c) x^2+b(c-a) x+c(a-b)=0\\ &\mathrm{x}=1 \text { is root } \therefore \text { other root is } 1\\ &\begin{aligned} & \alpha+\beta=-\frac{b(c-a)}{a(b-c)}=2 \\ & \Rightarrow-b c+a b=2 a b-2 a c \\ & \Rightarrow 2 a c=a b+b c \\ & \Rightarrow 2 a c=b(a+c) \\ & \Rightarrow 2 \mathrm{ac}=15 \mathrm{~b} \ldots(1) \\ & \Rightarrow 2 \mathrm{ac}=15\left(\frac{36}{5}\right)=108 \\ & \Rightarrow \mathrm{ac}=54 \\ & \mathrm{a}+\mathrm{c}=15 \\ & \mathrm{a}^2+\mathrm{c}^2+2 \mathrm{ac}=225 \\ & \mathrm{a}^2+\mathrm{c}^2=225-108=117 \end{aligned} \end{aligned}$$