`3x + 4y = 12 sqrt2` is the tangent to the ellipse `x^2/a^2 + y^2/9 = 1` then the distance between focii of ellipse is-

Options

A. 1. `2sqrt(7)`
B. 2. 4
C. 3. `2sqrt(5)`
D. 4. `2sqrt(2)`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Write the equation of the tangent line
The tangent line is given as:
\[ 3x + 4y = 12\sqrt{2} \]

### Step 2: Convert the tangent line to slope-intercept form
To convert the equation into the slope-intercept form \( y = mx + c \):
\[ 4y = -3x + 12\sqrt{2} \]
\[ y = -\frac{3}{4}x + 3\sqrt{2} \]
From this, we can identify:
- Slope \( m = -\frac{3}{4} \)
- Intercept \( c = 3\sqrt{2} \)

### Step 3: Identify the parameters of the ellipse
The equation of the ellipse is given as:
\[ \frac{x^2}{a^2} + \frac{y^2}{9} = 1 \]
Here, we have:
- \( b^2 = 9 \) (thus \( b = 3 \))
- \( a^2 = a^2 \) (we need to find \( a^2 \))

### Step 4: Use the tangency condition
For the line to be tangent to the ellipse, the following condition must hold:
\[ c^2 = a^2 m^2 + b^2 \]
Substituting the values we have:
\[ (3\sqrt{2})^2 = a^2 \left(-\frac{3}{4}\right)^2 + 9 \]
Calculating \( c^2 \):
\[ 18 = a^2 \cdot \frac{9}{16} + 9 \]

### Step 5: Solve for \( a^2 \)
Rearranging the equation:
\[ 18 - 9 = a^2 \cdot \frac{9}{16} \]
\[ 9 = a^2 \cdot \frac{9}{16} \]
Multiplying both sides by \( \frac{16}{9} \):
\[ a^2 = 16 \]

### Step 6: Calculate the eccentricity \( e \)
The eccentricity \( e \) of the ellipse is given by:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values:
\[ e = \sqrt{1 - \frac{9}{16}} = \sqrt{\frac{16 - 9}{16}} = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4} \]

### Step 7: Find the distance between the foci
The foci of the ellipse are located at \( (\pm ae, 0) \):
\[ ae = 4 \cdot \frac{\sqrt{7}}{4} = \sqrt{7} \]
Thus, the foci are at \( (\sqrt{7}, 0) \) and \( (-\sqrt{7}, 0) \).

### Step 8: Calculate the distance between the foci
The distance between the foci is given by:
\[ \text{Distance} = 2ae = 2\sqrt{7} \]

### Final Answer
The distance between the foci of the ellipse is:
\[ \boxed{2\sqrt{7}} \]