If ` a + b = 10 and sqrt((a)/(b)) - 13= - sqrt((b)/(a)) - 11 ` then what is the value of ` 3ab + 4a^(2) + 5 b ^(2)` ?

Options

A. (a) 450
B. (b) 300
C. (c) 600
D. (d) 750

Answer

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

1. **Given Equations:**
\[ a + b = 10 \]
\[ \sqrt{\frac{a}{b}} - 13 = -\sqrt{\frac{b}{a}} - 11 \]

2. **Rearranging the second equation:**
Let's simplify the second equation:
\[ \sqrt{\frac{a}{b}} + \sqrt{\frac{b}{a}} = 2 \]

3. **Substituting \(x = \sqrt{\frac{a}{b}}\):**
Then, we can express \(\sqrt{\frac{b}{a}}\) in terms of \(x\):
\[ \sqrt{\frac{b}{a}} = \frac{1}{x} \]
The equation becomes:
\[ x + \frac{1}{x} = 2 \]

4. **Multiplying through by \(x\):**
\[ x^2 + 1 = 2x \]
Rearranging gives:
\[ x^2 - 2x + 1 = 0 \]
This factors to:
\[ (x - 1)^2 = 0 \]
Thus, \(x = 1\).

5. **Finding values of \(a\) and \(b\):**
Since \(x = \sqrt{\frac{a}{b}} = 1\), we have:
\[ \frac{a}{b} = 1 \implies a = b \]
From the first equation \(a + b = 10\):
\[ a + a = 10 \implies 2a = 10 \implies a = 5 \]
Therefore, \(b = 5\).

6. **Calculating \(3ab + 4a^2 + 5b^2\):**
Now substituting \(a\) and \(b\) into the expression:
\[ 3ab + 4a^2 + 5b^2 = 3(5)(5) + 4(5^2) + 5(5^2) \]
Calculating each term:
\[ 3ab = 3 \cdot 5 \cdot 5 = 75 \]
\[ 4a^2 = 4 \cdot 5^2 = 4 \cdot 25 = 100 \]
\[ 5b^2 = 5 \cdot 5^2 = 5 \cdot 25 = 125 \]

7. **Adding the results:**
Now, adding all these values together:
\[ 75 + 100 + 125 = 300 \]

8. **Final Result:**
Thus, the value of \(3ab + 4a^2 + 5b^2\) is:
\[ \boxed{300} \]