A reactant (A) forms two products:
$A\xrightarrow{{{k_1}}}B$ , Activation energy ${E_{a1}}$ and pre-exponential factor A
$A\xrightarrow{{k2}}B$ , Activation energy \[{E_{a2}}\] and pre-exponential factor A
If ${E_{a1}}$ = 2\[{E_{a2}}\] , then ${k_1}$ and ${k_2}$ will be related as:
A. ${k_2} = {k_1}{e^{ - {E_{a1}}/RT}}$
B. ${k_2} = {k_1}{e^{ - {E_{a2}}/RT}}$
C. ${k_1} = A{k_2}{e^{ - {E_{a1/RT}}}}$
D. ${k_1} = A{k_2}{e^{ - {E_{a2/RT}}}}$

Hint: The rate of a reaction depends on the concentration of the reactants and the products that participate in the reaction process. When a reaction goes from reactant to product, the reaction is carried on the basis of the reaction coefficient. The rate of the reaction and the energy needed for the reaction can be related.

Complete step by step answer:
Just like the speed of a vehicle is defined by the distance covered by it in a specific amount of time, the rate of reaction can also be viewed as the change in concentration in time. Rate of reaction is defined as the change in reaction of reactants or products in unit time.
The rate of the reaction can be expressed as the decrease in the concentration of the involved reactants or the rate of increase in the product concentration in the reaction.
Considering the above reaction $A + B \to Products$ we can see that the concentration of A and B will be decreased while the product concentration is bound to be increased. The rate of the expression can be expressed in the terms of either of the two.
The representation of the rate of reaction in terms of the concentration of the reactants is known as rate law. Thus we can use the rate law to form a rate expression for the reaction.
The rate expression for the above reaction can be written as
$Rate = k[A][B]$
Here $k$ represents the rate constant and square brackets represent the molar concentration.
The rate constant can be related to the activation energy of the compound as
${k_2} = A{e^{ - {E_{a2}}/RT}}$
${k_1} = A{e^{ - {E_{a1}}/RT}}$
So we will get from the relations above that
$\dfrac{{{k_2}}}{{{k_1}}} = {e^{ - {E_{a1}}/RT}}$
And we get that because we have the relation ${E_{a2}} = 2{E_{a1}}$
So, ${k_2} = {k_1}{e^{ - {E_{a1}}/RT}}$

So, the correct answer is Option A.

Note: The reactions that occur in the chemistry need a certain energy for the procedure in the required effect. Reactions move forward if the required energy is given to the compounds. The activation energy acts like a threshold energy that is needed for the continuation of the reaction.