Question

The A.M. of two numbers is 34 and G.M is 16,the numbers are
A.2 and 64
B.64 and 3
C.64 and 4
D.None of these

Answer 1

Hint: Using the formulas of arithmetic mean and geometric mean form a quadratic equation , whose solution gives us the answer.

Complete step-by-step answer:
Step 1 :
Let the two numbers be a and b
We are given that the arithmetic mean is 34.
We know that the formula of arithmetic mean is $\dfrac{{a + b}}{2}$
Therefore, $\dfrac{{a + b}}{2} = 34$
Cross multiplying we get,
$ \Rightarrow a + b = 68$…………………..(1)
Step 2 :
We are given that the geometric of the two numbers is 16
We know that the formula of geometric mean is $\sqrt {ab} $
Therefore , $\sqrt {ab} = 16$……………(2)
Step 3 :
Now consider equation (1)
$ \Rightarrow a + b = 68$
Lets rewrite this equation as $a = 68 - b$…………. (3)
Step 4:
Squaring equation two on both sides we get,
$ab = 256$
Lets substitute $a = 68 - b$ in the above equation
$\begin{gathered}
   \Rightarrow b(68 - b) = 256 \\
   \Rightarrow 68b - {b^2} = 256 \\
\end{gathered} $
Rearranging the above equation we get,
$ \Rightarrow {b^2} - 68b + 256 = 0$
Now we have quadratic equation in b
Step 5 :
We need to solve the quadratic equation to find our answer.
Here let's use the method of factorization to solve the equation
We need to split the middle term of the equation (i.e.) – 68b into two whose product should be $256{b^2}$
Therefore - 68 b can be written as $ - 64b - 4b$
Hence the equation becomes
$ \Rightarrow {b^2} - 64b - 4b + 256 = 0$
Now let's take b common in the first two terms and – 4 common in the next two terms
$
   \Rightarrow b(b - 64) - 4(b - 64) = 0 \\
   \Rightarrow (b - 64)(b - 4) = 0 \\
    \\
$
From this we get that b=64 and 4
Step 6:
Now lets substitute the value of b in equation (1)
$\dfrac{{a + b}}{2} = 34$
$ \Rightarrow a + b = 68$
When b=64
\[
   \Rightarrow a + 64 = 68 \\
   \Rightarrow a = 68 - 64 \\
   \Rightarrow a = 4 \\
\]
When b = 4
\[
   \Rightarrow a + 4 = 68 \\
   \Rightarrow a = 68 - 4 \\
   \Rightarrow a = 64 \\
\]
Therefore , the values of a are 4 and 64
Therefore the two numbers are 4 and 64
The correct option is C

Note: In order to solve the quadratic equation we can even use other methods like formula method or completing the square method but it takes more time