Question

If x and y are positive integers and ${{x}^{2}}+{{y}^{2}}=1800$, then maximum value of x+y is?
A. 60
B. 52
C. 64
D. 48

Answer 1

Hint: Before solving this problem, You need to know the relation $\text{A}\text{.M}\text{.}\ge \text{G}\text{.M}\text{.}$ to get to the answer in the shortest possible manner. But before applying the relation convert the equation ${{x}^{2}}+{{y}^{2}}=1800$ to suitable form by using some basic algebraic formula and find the maximum value of x+y.

Complete step-by-step answer:
Given,
${{x}^{2}}+{{y}^{2}}=1800...........(i)$
To maximise: x + y
Now, we know:
${{\left( a+b \right)}^{2}}={{b}^{2}}+{{a}^{2}}+2ab$
Rearranging to suitable form;
${{\left( a+b \right)}^{2}}-2ab={{b}^{2}}+{{a}^{2}}$
Using the mentioned formula in equation (i) we get;
${{x}^{2}}+{{y}^{2}}=1800$
$\Rightarrow {{\left( x+y \right)}^{2}}-2xy=1800$
$\Rightarrow {{\left( x+y \right)}^{2}}=1800+2xy$
We know;
$\sqrt{{{k}^{2}}}=\pm k$
Applying;
\[x+y=\pm \sqrt{1800+2xy}\]
Now, as x and y are positive there sum must also be positive:
\[\Rightarrow x+y=\sqrt{1800+2xy}.....................(ii)\]
Now, we have to maximise x+y.
And from equation (ii) it is clear that for x+y to be maximum, xy should be maximum.
So, using the relation $\text{A}\text{.M}\text{.}\ge \text{G}\text{.M}\text{.}$ we get;
A.M. of x and y = $\dfrac{x+y}{2}$
G.M. of x and y= $\sqrt{xy}$
$\therefore \dfrac{x+y}{2}\ge \sqrt{xy}$
And for xy to be maximum, $\sqrt{xy}$ must be maximum.
So, G.M. is maximum when it is equal to A.M. and it is possible only if x and y are equal.
Therefore, we can say that x+y is maximum when x and y are equal.
Using this result in equation (i);
${{x}^{2}}+{{y}^{2}}=1800$
$\Rightarrow {{x}^{2}}+{{x}^{2}}=1800$
$\Rightarrow 2{{x}^{2}}=1800$
$\Rightarrow {{x}^{2}}=900$
$\Rightarrow x=\pm \sqrt{900}$
$\Rightarrow x=\pm \text{ }30$
And it is mentioned that x is positive so, x = 30.
$x=y=30$ for maximum value of x+y.
Therefore, maximum value of x+y is $30+30=60$
Hence, the answer is option A) 60.

Note: This method is only possible to use when all the variables involved in the relation of $\text{A}\text{.M}\text{.}\ge \text{G}\text{.M}\text{.}$are positive and the relation can be further extended as $\text{A}\text{.M}\text{.}\ge \text{G}\text{.M}\text{.}\ge \text{H}\text{.M}\text{.}$ . In such questions you can also give a try to the method of Derivatives for maximising an expression as well. However, that might get a bit complicated but if you can solve that you will surely get the answer.