Find the result after multiplication of $\left( x+7y \right)$ and $\left( 7x-y \right)$ .
(a) $7{{x}^{2}}+48xy-7{{y}^{2}}$
(b) $7{{x}^{2}}+48xy+7{{y}^{2}}$
(c) $7{{x}^{2}}-48xy-7{{y}^{2}}$
(d) $-7{{x}^{2}}-48xy+7{{y}^{2}}$

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Hint: For solving this question we will directly apply one formula and do the multiplication of the corresponding terms with each other and then solve for the correct answer.

Complete step-by-step answer:
We have to find the result of $\left( x+7y \right)\times \left( 7x-y \right)$ . Where $x$ and $y$ are variables.
Now, we will use the formula written below in this question to get the correct answer. We know that we can multiply any two variables with each other. Now, look at the formula given below:
$\left( a+b \right)\times \left( c+d \right)=ac+ad+bc+bd...........\left( 1 \right)$ .
Now, we will substitute $a=x$ , $b=7y$ , $c=7x$ and $d=-y$ . Then,
  & \left( x+7y \right)\times \left( 7x-y \right) \\
 & \Rightarrow 7{{x}^{2}}-xy+49xy-7{{y}^{2}} \\
 & \Rightarrow 7{{x}^{2}}+48xy-7{{y}^{2}} \\

Thus, from the above result, we can say that, $\left( x+7y \right)\times \left( 7x-y \right)=7{{x}^{2}}+48xy-7{{y}^{2}}$ . Now, we can select the correct option from the given options. Then, we can say that (a) is the correct option.
Hence, option (a) is the correct option.

Note: The question is very easy to solve and we can solve it without any formulae also, where we can easily do the calculations just like we multiply any two numerical values we can multiply variables with one another and then do the arithmetic calculations like we can add and subtract the coefficient of same variables as per their sign. One more important point is that the student should take care of any calculation mistake while solving the question to get the correct answer.