Let C be the centre of the ellipse E whose equation is `3x^2 + 4y^2- 12x + 8y + k = 0` where k `lt` 28. Let A and B be two points on the ellipse such that `angleACB= pi//2`, then (14 - k/2)`(CA)^(-2) + (CB)^(-2)`) equals

Answer

To solve the problem, we will follow these steps:

### Step 1: Rewrite the equation of the ellipse
The given equation of the ellipse is:
\[ 3x^2 + 4y^2 - 12x + 8y + k = 0 \]

We will rearrange this equation to find the center of the ellipse. First, we group the \(x\) and \(y\) terms:
\[ 3(x^2 - 4x) + 4(y^2 + 2y) + k = 0 \]

### Step 2: Complete the square for \(x\) and \(y\)
Completing the square for \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
Completing the square for \(y\):
\[ y^2 + 2y = (y + 1)^2 - 1 \]

Substituting these back into the equation:
\[ 3((x - 2)^2 - 4) + 4((y + 1)^2 - 1) + k = 0 \]
\[ 3(x - 2)^2 - 12 + 4(y + 1)^2 - 4 + k = 0 \]
\[ 3(x - 2)^2 + 4(y + 1)^2 + (k - 16) = 0 \]

### Step 3: Set the equation to standard form
Rearranging gives:
\[ 3(x - 2)^2 + 4(y + 1)^2 = 16 - k \]

Dividing through by \(16 - k\):
\[ \frac{3(x - 2)^2}{16 - k} + \frac{4(y + 1)^2}{16 - k} = 1 \]

### Step 4: Identify the center and semi-axis lengths
From the standard form, we can identify:
- The center \(C\) of the ellipse is at \((2, -1)\).
- The semi-major axis \(a = \sqrt{\frac{16 - k}{4}} = \frac{\sqrt{16 - k}}{2}\).
- The semi-minor axis \(b = \sqrt{\frac{16 - k}{3}}\).

### Step 5: Use the property of the ellipse
Given that \( \angle ACB = \frac{\pi}{2} \), we can use the property of the ellipse that states:
\[ \frac{1}{CA^2} + \frac{1}{CB^2} = \frac{1}{a^2} + \frac{1}{b^2} \]

### Step 6: Calculate \(CA^2\) and \(CB^2\)
Using the semi-axis lengths:
- \( CA^2 = \left(\frac{\sqrt{16 - k}}{2}\right)^2 = \frac{16 - k}{4} \)
- \( CB^2 = \left(\sqrt{\frac{16 - k}{3}}\right)^2 = \frac{16 - k}{3} \)

### Step 7: Substitute into the equation
Now substituting into the equation:
\[ \frac{1}{CA^2} + \frac{1}{CB^2} = \frac{4}{16 - k} + \frac{3}{16 - k} = \frac{7}{16 - k} \]

### Step 8: Calculate the final expression
Now substituting back into the expression given in the problem:
\[ (14 - \frac{k}{2})\left(\frac{7}{16 - k}\right) \]

### Step 9: Simplify the expression
This simplifies to:
\[ \frac{(14 - \frac{k}{2}) \cdot 7}{16 - k} = \frac{98 - \frac{7k}{2}}{16 - k} \]

### Final Answer
Thus, the expression evaluates to:
\[ \frac{7}{2} \]