An ellipse whose distance between foci `S and S'` is 4 units is inscribed in the `DeltaABC` touching thesides `AB, AC and BC at P, Q and R`. If centre of ellipse is at origin and major axis along x-axis `SP + S'P = 6`, then

Options

A. `9x^(2)+5y^(2)=45`
B. `4x^(2)+9y^(2)=46`
C. `5x^(2)+9y^(2)=45`
D. `9x^(2)+4y^(2)=36`

Answer

To solve the problem, we need to find the equation of the ellipse inscribed in triangle ABC, given the conditions. Let's break it down step by step.

### Step 1: Understanding the Given Information
- The distance between the foci \( S \) and \( S' \) is 4 units.
- The center of the ellipse is at the origin (0, 0).
- The major axis is along the x-axis.
- The sum of the distances from the foci to a point \( P \) on the ellipse is given as \( SP + S'P = 6 \).

### Step 2: Relating the Distance Between Foci to the Major Axis
The distance between the foci \( S \) and \( S' \) is given by \( 2c \), where \( c \) is the distance from the center to each focus. Since the distance is 4 units, we have:
\[ 2c = 4 \implies c = 2 \]

### Step 3: Using the Relationship Between Major Axis and Foci
For an ellipse, the relationship between the semi-major axis \( a \), semi-minor axis \( b \), and the distance \( c \) is given by:
\[ c^2 = a^2 - b^2 \]
We also know from the property of ellipses that:
\[ SP + S'P = 2a \]
Given \( SP + S'P = 6 \), we can equate this to find \( a \):
\[ 2a = 6 \implies a = 3 \]

### Step 4: Finding the Value of \( b \)
Now we have \( a = 3 \) and \( c = 2 \). We can use the relationship \( c^2 = a^2 - b^2 \):
\[ c^2 = 2^2 = 4 \]
\[ a^2 = 3^2 = 9 \]
Substituting these values into the equation:
\[ 4 = 9 - b^2 \implies b^2 = 9 - 4 = 5 \]

### Step 5: Writing the Equation of the Ellipse
The standard form of the equation of an ellipse centered at the origin with the major axis along the x-axis is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ \frac{x^2}{9} + \frac{y^2}{5} = 1 \]

### Step 6: Rearranging the Equation
To express this in a more standard form, we can multiply through by 45 (the least common multiple of the denominators):
\[ 5x^2 + 9y^2 = 45 \]

### Final Answer
The equation of the ellipse is:
\[ 5x^2 + 9y^2 = 45 \]