The axes are rotated through an angle, keeping origin fixed then the equation of the line `3x+4y-12=0` becomes `ax+by-ab=0`.If `1/(a^(2))+(1)/(b^(2))=k^(2).` then the value of 12k is

Answer

To solve the problem step by step, we will follow the process outlined in the video transcript.

### Step 1: Understand the given equation
We start with the line equation:
\[ 3x + 4y - 12 = 0 \]

### Step 2: Find the distance from the origin to the line
The distance \(d\) from the origin (0, 0) to the line \(Ax + By + C = 0\) is given by the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
For our line:
- \(A = 3\)
- \(B = 4\)
- \(C = -12\)

Substituting these values into the formula:
\[ d = \frac{|3(0) + 4(0) - 12|}{\sqrt{3^2 + 4^2}} = \frac{|-12|}{\sqrt{9 + 16}} = \frac{12}{\sqrt{25}} = \frac{12}{5} \]

### Step 3: Set up the new line equation after rotation
After rotating the axes, the line transforms to:
\[ ax + by - ab = 0 \]

### Step 4: Find the distance from the origin to the new line
Using the same distance formula for the new line:
\[ d' = \frac{|a(0) + b(0) - ab|}{\sqrt{a^2 + b^2}} = \frac{| - ab |}{\sqrt{a^2 + b^2}} = \frac{ab}{\sqrt{a^2 + b^2}} \]

### Step 5: Set the distances equal
Since the distances from the origin to both lines must be equal:
\[ \frac{12}{5} = \frac{ab}{\sqrt{a^2 + b^2}} \]

### Step 6: Cross-multiply and simplify
Cross-multiplying gives:
\[ 12\sqrt{a^2 + b^2} = 5ab \]
Squaring both sides:
\[ 144(a^2 + b^2) = 25a^2b^2 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ 25a^2b^2 - 144a^2 - 144b^2 = 0 \]

### Step 8: Divide by \(a^2b^2\)
Dividing through by \(a^2b^2\):
\[ 25 - \frac{144}{b^2} - \frac{144}{a^2} = 0 \]

### Step 9: Rearranging to find \( \frac{1}{a^2} + \frac{1}{b^2} \)
This can be rearranged to:
\[ \frac{144}{a^2} + \frac{144}{b^2} = 25 \]
Dividing through by 144:
\[ \frac{1}{a^2} + \frac{1}{b^2} = \frac{25}{144} \]

### Step 10: Relate to \(k^2\)
From the problem, we know:
\[ \frac{1}{a^2} + \frac{1}{b^2} = k^2 \]
Thus, we have:
\[ k^2 = \frac{25}{144} \]

### Step 11: Solve for \(k\)
Taking the square root gives:
\[ k = \frac{5}{12} \]

### Step 12: Find \(12k\)
Finally, we calculate:
\[ 12k = 12 \times \frac{5}{12} = 5 \]

### Final Answer
Thus, the value of \(12k\) is:
\[ \boxed{5} \]