If the equation $\mathrm{a}(\mathrm{b}-\mathrm{c}) \mathrm{x}^2+\mathrm{b}(\mathrm{c}-\mathrm{a}) \mathrm{x}+\mathrm{c}(\mathrm{a}-\mathrm{b})=0$ has equal roots, where $\mathrm{a}+\mathrm{c}=15$ and $\mathrm{b}=\frac{36}{5}$, then $a^2+c^2$ is equal to

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Aicharya · Accepted Answer

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

If the equation $\mathrm{a}(\mathrm{b}-\mathrm{c}) \mathrm{x}^2+\mathrm{b}(\mathrm{c}-\mathrm{a})…

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