Question

If $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f}$ then show that $({b^2} + {d^2} + {f^2}) \times ({a^2} + {c^2} + {e^2}) = {(ab + cd + ef)^2}$
A. True
B. False

Answer 1

Hint: In order to solve this question we solve both the sides of the equation and prove them equal to one another. So, firstly we will consider the constant $k$ which is equal to the terms $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f}$ and on further solving these terms into one terms of another and then substitute the values of the terms in both side of the equation and on solving further we get our required result.

Complete step by step answer:
Here, we have to show that $({b^2} + {d^2} + {f^2}) \times ({a^2} + {c^2} + {e^2}) = {(ab + cd + ef)^2}$
Given that $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f}$
Let us consider $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = k$ then
$ \Rightarrow a = bk,\,\,c = dk,\,\,e = fk$
Now, consider the left side of the equation i.e., $({b^2} + {d^2} + {f^2}) \times ({a^2} + {c^2} + {e^2})$
Substituting the value of $a,\,b$ and $c$. We get,
$ \Rightarrow ({b^2} + {d^2} + {f^2})({b^2}{k^2} + {d^2}{k^2} + {f^2}{k^2})$
Taking ${k^2}$ as a common factor from the equation. We get,
$ \Rightarrow {k^2}({b^2} + {d^2} + {f^2})({b^2} + {d^2} + {f^2})$

On multiplying we get,
$ \Rightarrow {k^2}{\left( {{b^2} + {d^2} + {f^2}} \right)^2}$
Now considering the right side of the equation i.e., ${(ab + cd + ef)^2}$
Substituting the value of $a,\,b$ and $c$. We get,
$ \Rightarrow {\left( {bk \cdot b + dk \cdot d + fk \cdot f} \right)^2}$
On multiplying we get,
$ \Rightarrow {\left( {{b^2}k + {d^2}k + {f^2}k} \right)^2}$
Taking ${k^2}$ as a common factor from the equation. We get,
$ \Rightarrow {k^2}{\left( {{b^2} + {d^2} + {f^2}} \right)^2}$
Therefore, the left side of the equation is equal to the right side of the equation.Hence, the above statement is True.

Therefore, the correct option is A.

Note: In these types of problems in which we have to prove left side of the equation is equal to the right side of the equation if we are unable to prove one side of the equation equal to the other side we can solve both side of the equation and prove their results equal to one another. Note that here we have consider a one constant $k$ as the terms $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f}$ are equal to so we have consider only one constant to solve this question.