The difference between two numbers is 18 and their HCF and LCM are 6 and 168 respectively. What is the sum of squares of the two numbers ?

Options

A. 2280
B. 2260
C. 2420
D. 2340

Answer

To solve the problem step-by-step, we will follow these steps:

### Step 1: Define the variables
Let the two numbers be \( x \) and \( y \). According to the problem, we know:
- The difference between the two numbers is given by:
\[ x - y = 18 \quad \text{(Equation 1)} \]

### Step 2: Use the HCF and LCM
We are also given the HCF (Highest Common Factor) and LCM (Least Common Multiple) of the two numbers:
- HCF = 6
- LCM = 168

We know that the product of two numbers can be expressed as:
\[ x \cdot y = \text{HCF} \times \text{LCM} \]
Substituting the values:
\[ x \cdot y = 6 \times 168 = 1008 \quad \text{(Equation 2)} \]

### Step 3: Express \( x \) in terms of \( y \)
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = y + 18 \]

### Step 4: Substitute \( x \) in Equation 2
Now, substitute \( x \) in Equation 2:
\[ (y + 18) \cdot y = 1008 \]
Expanding this gives:
\[ y^2 + 18y - 1008 = 0 \]

### Step 5: Solve the quadratic equation
We can solve the quadratic equation using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 18 \), and \( c = -1008 \):
\[ y = \frac{-18 \pm \sqrt{18^2 - 4 \cdot 1 \cdot (-1008)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 18^2 = 324 \]
\[ -4 \cdot 1 \cdot (-1008) = 4032 \]
\[ b^2 - 4ac = 324 + 4032 = 4356 \]
Now, take the square root:
\[ \sqrt{4356} = 66 \]
Now substituting back into the formula:
\[ y = \frac{-18 \pm 66}{2} \]
Calculating the two possible values for \( y \):
1. \( y = \frac{48}{2} = 24 \)
2. \( y = \frac{-84}{2} = -42 \) (not valid since \( y \) must be positive)

So, \( y = 24 \).

### Step 6: Find \( x \)
Now substitute \( y \) back to find \( x \):
\[ x = y + 18 = 24 + 18 = 42 \]

### Step 7: Calculate the sum of squares
Now, we need to find the sum of squares of the two numbers:
\[ x^2 + y^2 = 42^2 + 24^2 \]
Calculating each square:
\[ 42^2 = 1764 \]
\[ 24^2 = 576 \]
Now, adding them together:
\[ x^2 + y^2 = 1764 + 576 = 2340 \]

### Final Answer
Thus, the sum of the squares of the two numbers is:
\[ \boxed{2340} \]