An ellipse has the points `(1, -1) and (2,-1)` as its foci and `x + y = 5` as one of its tangent then the value of `a^2+b^2` where `a,b` are the lenghta of semi major and minor axes of ellipse respectively is :

Options

A. `(41)/(2)`
B. 10
C. 19
D. `(81)/(4)`

Explanation

To solve the problem step by step, we will follow the information provided in the question and derive the necessary values.

### Step 1: Identify the coordinates of the foci
The foci of the ellipse are given as the points (1, -1) and (2, -1). We can denote the coordinates of the foci as:
- F1 = (1, -1)
- F2 = (2, -1)

### Step 2: Find the center of the ellipse
The center (α, β) of the ellipse is the midpoint of the line segment joining the foci. We can calculate it as follows:
\[ \alpha = \frac{x_1 + x_2}{2} = \frac{1 + 2}{2} = \frac{3}{2} \]
\[ \beta = \frac{y_1 + y_2}{2} = \frac{-1 + (-1)}{2} = -1 \]
Thus, the center of the ellipse is (3/2, -1).

### Step 3: Write the standard form of the ellipse
The standard form of the ellipse with center (α, β) is given by:
\[ \frac{(x - \alpha)^2}{a^2} + \frac{(y - \beta)^2}{b^2} = 1 \]
Substituting the values of α and β, we have:
\[ \frac{(x - \frac{3}{2})^2}{a^2} + \frac{(y + 1)^2}{b^2} = 1 \]

### Step 4: Determine the distance between the foci
The distance between the foci is given by:
\[ c = \frac{d}{2} = \frac{2 - 1}{2} = \frac{1}{2} \]
In an ellipse, we have the relationship:
\[ c^2 = a^2 - b^2 \]
Thus, we have:
\[ c = \frac{1}{2} \implies c^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 5: Write the equation of the tangent
The equation of the tangent is given as \(x + y = 5\). We can rewrite this as:
\[ y = -x + 5 \]
The slope (m) of this line is -1.

### Step 6: Use the tangent equation for the ellipse
The general equation of the tangent to the ellipse at point (x_0, y_0) is given by:
\[ y + 1 = m(x - \frac{3}{2}) \pm \sqrt{a^2 m^2 + b^2} \]
Substituting \(m = -1\):
\[ y + 1 = -1(x - \frac{3}{2}) \pm \sqrt{a^2 + b^2} \]
This simplifies to:
\[ y + 1 = -x + \frac{3}{2} \pm \sqrt{a^2 + b^2} \]
Comparing this with \(y = -x + 5\), we set the constant terms equal:
\[ 5 = -\frac{3}{2} \pm \sqrt{a^2 + b^2} - 1 \]
Thus:
\[ 5 + 1 + \frac{3}{2} = \sqrt{a^2 + b^2} \]
\[ 6.5 = \sqrt{a^2 + b^2} \]

### Step 7: Solve for \(a^2 + b^2\)
Squaring both sides:
\[ (6.5)^2 = a^2 + b^2 \]
\[ 42.25 = a^2 + b^2 \]

### Step 8: Final result
To express it as a fraction:
\[ 42.25 = \frac{169}{4} \]
Thus, the value of \(a^2 + b^2\) is:
\[ \boxed{\frac{169}{4}} \]