If `4x^(2) + y^(2) = 40` and `xy = 6, (x gt , y gt 0)` then the value of `2x + y` is

Options

A. 4
B. 24
C. 8
D. 16

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations**:
- \( 4x^2 + y^2 = 40 \) (Equation 1)
- \( xy = 6 \) (Equation 2)

2. **Finding \( 2x + y \)**:
We want to find the value of \( 2x + y \). To do this, we can use the identity:
\[ (2x + y)^2 = (2x)^2 + y^2 + 2(2x)(y) \]
This simplifies to:
\[ (2x + y)^2 = 4x^2 + y^2 + 4xy \]

3. **Substituting Values**:
Now, we substitute the values from Equation 1 and Equation 2 into the identity:
- From Equation 1, we have \( 4x^2 + y^2 = 40 \).
- From Equation 2, we have \( xy = 6 \), so \( 4xy = 4 \times 6 = 24 \).

Substituting these values into the equation gives:
\[ (2x + y)^2 = 40 + 24 \]

4. **Calculating**:
Now we calculate:
\[ (2x + y)^2 = 64 \]

5. **Taking the Square Root**:
To find \( 2x + y \), we take the square root of both sides:
\[ 2x + y = \sqrt{64} \]
Since \( x > 0 \) and \( y > 0 \), we only consider the positive root:
\[ 2x + y = 8 \]

Thus, the value of \( 2x + y \) is \( 8 \).