If `a/sqrt(b c)-2=sqrt(b/c)+sqrt(c/b),` where `a , b , c >0,` then the family of lines `sqrt(a)x+sqrt(b)y+sqrt(c)=0` passes though the fixed point given by (a)`(1,1)` (b) `(1,-2)` (c)`(-1,2)` (d) `(-1,1)`

Options

A. (1,1)
B. (1,-2)
C. (-1,2)
D. (-1,1)

Explanation

To solve the problem, we start with the given equation:

\[ \frac{a}{\sqrt{bc}} - 2 = \sqrt{\frac{b}{c}} + \sqrt{\frac{c}{b}} \]

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate \(a\):

\[ \frac{a}{\sqrt{bc}} = 2 + \sqrt{\frac{b}{c}} + \sqrt{\frac{c}{b}} \]

### Step 2: Finding a Common Denominator
Next, we can find a common denominator for the right-hand side. The common denominator for \(\sqrt{bc}\) is \(\sqrt{bc}\):

\[ \frac{a}{\sqrt{bc}} = 2 + \frac{\sqrt{b^2} + \sqrt{c^2}}{\sqrt{bc}} = 2 + \frac{\sqrt{b} \cdot \sqrt{c} + \sqrt{c} \cdot \sqrt{b}}{\sqrt{bc}} \]

### Step 3: Simplifying the Right Side
This simplifies to:

\[ \frac{a}{\sqrt{bc}} = 2 + \frac{2\sqrt{bc}}{\sqrt{bc}} = 2 + 2 = 4 \]

### Step 4: Solving for \(a\)
Now, we can solve for \(a\):

\[ a = 4\sqrt{bc} \]

### Step 5: Analyzing the Family of Lines
The family of lines is given by:

\[ \sqrt{a}x + \sqrt{b}y + \sqrt{c} = 0 \]

Substituting \(a = 4\sqrt{bc}\):

\[ \sqrt{4\sqrt{bc}}x + \sqrt{b}y + \sqrt{c} = 0 \]

This simplifies to:

\[ 2(\sqrt[4]{b}\sqrt[4]{c})x + \sqrt{b}y + \sqrt{c} = 0 \]

### Step 6: Finding the Fixed Point
To find the fixed point, we can substitute the coordinates of the options into the equation. Let's check option (d) \((-1, 1)\):

Substituting \(x = -1\) and \(y = 1\):

\[ 2(\sqrt[4]{b}\sqrt[4]{c})(-1) + \sqrt{b}(1) + \sqrt{c} = 0 \]

This gives:

\[ -2\sqrt[4]{bc} + \sqrt{b} + \sqrt{c} = 0 \]

Rearranging gives:

\[ \sqrt{b} + \sqrt{c} = 2\sqrt[4]{bc} \]

### Step 7: Verifying the Condition
This condition is satisfied if we apply the identity \(x^2 + y^2 \geq 2xy\) (the Cauchy-Schwarz inequality), which holds true for positive \(b\) and \(c\). Thus, the line passes through the fixed point \((-1, 1)\).

### Conclusion
The family of lines passes through the fixed point:

\[ \text{Answer: } (-1, 1) \]