If  a² + 4b² + 25c² = 2ab + 10bc + 5ca , prove that a = 2b = 5c

Answer :

We have  :  a² + 4 b² + 25 c² =  2 ab  + 10 bc  + 5 ca

And to show a  =  2b  =  5c  ,

a  =  2b 

b  = $\frac{a}{2}$
And

a  = 5c

c  = $\frac{a}{5}$

Now we substitute these values of b and c in given equation and get

$$\begin{gathered} \Rightarrow a^2 + 4{\left(\frac{a}{2}\right)}^2 + 25{\left(\frac{a}{5}\right)}^2 = 2 \times a \times \frac{a}{2} +\hspace{0.17em}10\times \frac{a}{2}\times \frac{a}{5} + 5 \times \frac{a}{5}\times a \\ \Rightarrow a^2 + 4\times {\frac{a}{4}}^2 + 25\times {\frac{a}{25}}^2= a^2 +\hspace{0.17em}{a}^2 +a^2 \\ \Rightarrow a^2 +a^2 + a^2= 3a^2 \\ \mathbf{\Rightarrow }\mathbf{3}\mathbf{ }{\mathit{a}}^{\mathbf{2}}\mathbf{=}\mathbf{ }\mathbf{3}{\mathit{a}}^{\mathbf{2}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathit{H}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathbf{ }\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{d}\mathbf{ }\mathbf{)} \end{gathered}$$