QJEEM25Ma964e: JEE Main 2025 Mathematics - Class 11 JEEM25Ma964e Solution: If the equation has equal roots, where and , then is equal to…

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