If `5a+(1)/(3a)=5`, then value of `9a^(2)+(1)/(25a^(2))` is

Options

A. `(51)/(5)`
B. `(29)/(5)`
C. `(52)/(5)`
D. `(39)/(5)`

Answer

To solve the equation \( 5a + \frac{1}{3a} = 5 \) and find the value of \( 9a^2 + \frac{1}{25a^2} \), we will follow these steps:

### Step 1: Simplify the given equation
We start with the equation:
\[ 5a + \frac{1}{3a} = 5 \]
To eliminate the fraction, we can multiply both sides by \( 3a \):
\[ 3a(5a) + 3a\left(\frac{1}{3a}\right) = 3a(5) \]
This simplifies to:
\[ 15a^2 + 1 = 15a \]

### Step 2: Rearrange the equation
Now, we rearrange the equation to set it to zero:
\[ 15a^2 - 15a + 1 = 0 \]

### Step 3: Use the quadratic formula
We can use the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the values of \( a \). Here, \( a = 15 \), \( b = -15 \), and \( c = 1 \):
\[ a = \frac{-(-15) \pm \sqrt{(-15)^2 - 4 \cdot 15 \cdot 1}}{2 \cdot 15} \]
Calculating the discriminant:
\[ (-15)^2 - 4 \cdot 15 \cdot 1 = 225 - 60 = 165 \]
Now substituting back into the formula:
\[ a = \frac{15 \pm \sqrt{165}}{30} \]
This gives us the values of \( a \).

### Step 4: Find \( 9a^2 + \frac{1}{25a^2} \)
Next, we need to find \( 9a^2 + \frac{1}{25a^2} \). To do this, we first find \( 3a + \frac{1}{5a} \) by squaring the earlier expression:
\[ (3a + \frac{1}{5a})^2 = 9a^2 + 2(3a)(\frac{1}{5a}) + \frac{1}{25a^2} \]
This simplifies to:
\[ 9a^2 + \frac{1}{25a^2} + \frac{6}{5} = 9 \]
So we rearrange to find:
\[ 9a^2 + \frac{1}{25a^2} = 9 - \frac{6}{5} \]
Calculating \( 9 - \frac{6}{5} \):
\[ 9 = \frac{45}{5} \quad \Rightarrow \quad 9 - \frac{6}{5} = \frac{45}{5} - \frac{6}{5} = \frac{39}{5} \]

### Final Answer
Thus, the value of \( 9a^2 + \frac{1}{25a^2} \) is:
\[ \frac{39}{5} \]