If `(4 sqrt(3) + 5 sqrt(2))/(sqrt(48 ) + sqrt(18))= a + bsqrt(6)` , then the value of a and b are respectively

Options

A. `(9)/(15),-(4)/(15)`
B. `(3)/(11),(4)/(33)`
C. `(9)/(10),(2)/(5)`
D. `(3)/(5),(4)/(15)`

Answer

To solve the equation

\[ \frac{4 \sqrt{3} + 5 \sqrt{2}}{\sqrt{48} + \sqrt{18}} = a + b \sqrt{6} \]

we will follow these steps:

### Step 1: Simplify the Denominator
First, we simplify the denominator \(\sqrt{48} + \sqrt{18}\).

\[ \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} \]
\[ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \]

Thus, the denominator becomes:

\[ \sqrt{48} + \sqrt{18} = 4\sqrt{3} + 3\sqrt{2} \]

### Step 2: Rationalize the Denominator
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(4\sqrt{3} - 3\sqrt{2}\):

\[ \frac{(4\sqrt{3} + 5\sqrt{2})(4\sqrt{3} - 3\sqrt{2})}{(4\sqrt{3} + 3\sqrt{2})(4\sqrt{3} - 3\sqrt{2})} \]

### Step 3: Calculate the Denominator
Using the difference of squares:

\[ (4\sqrt{3})^2 - (3\sqrt{2})^2 = 48 - 18 = 30 \]

### Step 4: Calculate the Numerator
Now we calculate the numerator:

\[ (4\sqrt{3})(4\sqrt{3}) - (4\sqrt{3})(3\sqrt{2}) + (5\sqrt{2})(4\sqrt{3}) - (5\sqrt{2})(3\sqrt{2}) \]
\[ = 16 \cdot 3 - 12\sqrt{6} + 20\sqrt{6} - 15 \cdot 2 \]
\[ = 48 - 30 + 8\sqrt{6} \]
\[ = 18 + 8\sqrt{6} \]

### Step 5: Combine the Results
Now we have:

\[ \frac{18 + 8\sqrt{6}}{30} \]

We can separate this into two fractions:

\[ = \frac{18}{30} + \frac{8\sqrt{6}}{30} \]
\[ = \frac{3}{5} + \frac{4\sqrt{6}}{15} \]

### Step 6: Identify \(a\) and \(b\)
From the expression \(\frac{3}{5} + \frac{4\sqrt{6}}{15}\), we can identify:

\[ a = \frac{3}{5}, \quad b = \frac{4}{15} \]

### Final Answer
Thus, the values of \(a\) and \(b\) are:

\[ \boxed{\left(\frac{3}{5}, \frac{4}{15}\right)} \]