If ` 3 sqrt(( 1 - a)/( a)) + 9 = 19 - 3 sqrt(( a)/( 1 - a))` then what is the value of a

Options

A. `(3)/(10) , (7)/(10)`
B. `(1)/(10),(9)/(10)`
C. `(2)/(5),(3)/(5)`
D. `(1)/(5),(4)/(5)`

Answer

To solve the equation \( 3 \sqrt{\frac{1 - a}{a}} + 9 = 19 - 3 \sqrt{\frac{a}{1 - a}} \), we will follow these steps:

### Step 1: Substitute a variable for the square roots
Let \( t = \sqrt{\frac{1 - a}{a}} \). Then, we can express \( \sqrt{\frac{a}{1 - a}} \) in terms of \( t \):
\[ \sqrt{\frac{a}{1 - a}} = \frac{1}{t} \]
Now, we can rewrite the equation:
\[ 3t + 9 = 19 - 3 \cdot \frac{1}{t} \]

### Step 2: Rearranging the equation
Rearranging gives us:
\[ 3t + 3 \cdot \frac{1}{t} = 19 - 9 \]
\[ 3t + \frac{3}{t} = 10 \]

### Step 3: Multiply through by \( t \) to eliminate the fraction
Multiplying both sides by \( t \) (assuming \( t \neq 0 \)):
\[ 3t^2 + 3 = 10t \]

### Step 4: Rearranging into standard quadratic form
Rearranging gives us:
\[ 3t^2 - 10t + 3 = 0 \]

### Step 5: Factor the quadratic equation
We can factor this quadratic equation:
\[ 3t^2 - 9t - t + 3 = 0 \]
\[ (3t - 1)(t - 3) = 0 \]

### Step 6: Solve for \( t \)
Setting each factor to zero gives us:
1. \( 3t - 1 = 0 \) → \( t = \frac{1}{3} \)
2. \( t - 3 = 0 \) → \( t = 3 \)

### Step 7: Substitute back to find \( a \)
Now we substitute back to find \( a \):
1. For \( t = 3 \):
\[ \sqrt{\frac{1 - a}{a}} = 3 \]
Squaring both sides:
\[ \frac{1 - a}{a} = 9 \implies 1 - a = 9a \implies 1 = 10a \implies a = \frac{1}{10} \]

2. For \( t = \frac{1}{3} \):
\[ \sqrt{\frac{1 - a}{a}} = \frac{1}{3} \]
Squaring both sides:
\[ \frac{1 - a}{a} = \frac{1}{9} \implies 1 - a = \frac{a}{9} \implies 9 - 9a = a \implies 9 = 10a \implies a = \frac{9}{10} \]

### Final Values of \( a \)
Thus, the possible values of \( a \) are:
\[ a = \frac{1}{10} \quad \text{and} \quad a = \frac{9}{10} \]