If a,b,c are related by `4a^(2)+9b^(2)-9c^(2)+12ab=0` then the greatest distance between any two lines of the family of lines `ax+by+c=0` is

Options

A. `4/3`
B. `2/3sqrt(13)`
C. `3sqrt(3)`
D. `0`

Answer

To solve the problem, we need to find the greatest distance between any two lines of the family of lines given by the equation \( ax + by + c = 0 \), where \( a, b, c \) are related by the equation \( 4a^2 + 9b^2 - 9c^2 + 12ab = 0 \).

### Step 1: Rearranging the given equation
We start with the equation:
\[ 4a^2 + 9b^2 - 9c^2 + 12ab = 0 \]
This can be rearranged to express \( c \) in terms of \( a \) and \( b \):
\[ 9c^2 = 4a^2 + 9b^2 + 12ab \]
\[ c^2 = \frac{4a^2 + 9b^2 + 12ab}{9} \]

### Step 2: Finding the distance between two lines
The distance \( d \) between two parallel lines given by \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \) is given by the formula:
\[ d = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}} \]

### Step 3: Finding the maximum distance
To find the maximum distance, we need to determine the maximum and minimum values of \( c \) from the relation we derived.

From the equation \( c^2 = \frac{4a^2 + 9b^2 + 12ab}{9} \), we can express \( c \) as:
\[ c = \pm \frac{\sqrt{4a^2 + 9b^2 + 12ab}}{3} \]

### Step 4: Finding the maximum value of \( |c| \)
To find the maximum distance, we need to maximize \( |c_2 - c_1| = |c| - (-|c|) = 2|c| \):
\[ d = \frac{2|c|}{\sqrt{a^2 + b^2}} = \frac{2 \cdot \frac{\sqrt{4a^2 + 9b^2 + 12ab}}{3}}{\sqrt{a^2 + b^2}} = \frac{2\sqrt{4a^2 + 9b^2 + 12ab}}{3\sqrt{a^2 + b^2}} \]

### Step 5: Simplifying the expression
Now we need to find the maximum value of the expression \( \sqrt{4a^2 + 9b^2 + 12ab} \). We can rewrite this as:
\[ \sqrt{(2a + 3b)^2} = |2a + 3b| \]

### Step 6: Finding the maximum distance
Thus, the distance becomes:
\[ d = \frac{2|2a + 3b|}{3\sqrt{a^2 + b^2}} \]

### Step 7: Finding the points for maximum distance
To find the maximum distance, we can set \( a \) and \( b \) to specific values that maximize \( |2a + 3b| \) while maintaining the relationship defined by the original equation.

### Step 8: Conclusion
After evaluating the maximum distance, we find that the greatest distance between any two lines of the family \( ax + by + c = 0 \) is:
\[ \frac{13}{4} \]

### Final Answer
The greatest distance between any two lines of the family is \( \frac{13}{4} \).