Let e be the eccentricity of the ellipse represented by x=5 `(2cos theta+3 sin theta)`, y =4 (2 sin `theta`-3 cos `theta`), then `e^2` equals

Answer

To find the value of \( e^2 \) for the ellipse represented by the parametric equations \( x = 5(2 \cos \theta + 3 \sin \theta) \) and \( y = 4(2 \sin \theta - 3 \cos \theta) \), we will follow these steps:

### Step 1: Rewrite the Parametric Equations
We start with the given parametric equations:
\[ x = 5(2 \cos \theta + 3 \sin \theta) \]
\[ y = 4(2 \sin \theta - 3 \cos \theta) \]

### Step 2: Normalize the Equations
Divide both equations by their respective coefficients:
\[ \frac{x}{5} = 2 \cos \theta + 3 \sin \theta \quad \text{(Equation 1)} \]
\[ \frac{y}{4} = 2 \sin \theta - 3 \cos \theta \quad \text{(Equation 2)} \]

### Step 3: Square and Add the Equations
Now, we square both equations and add them:
\[ \left(\frac{x}{5}\right)^2 + \left(\frac{y}{4}\right)^2 = (2 \cos \theta + 3 \sin \theta)^2 + (2 \sin \theta - 3 \cos \theta)^2 \]

### Step 4: Expand the Right Side
Expanding the right side:
\[ (2 \cos \theta + 3 \sin \theta)^2 = 4 \cos^2 \theta + 12 \cos \theta \sin \theta + 9 \sin^2 \theta \]
\[ (2 \sin \theta - 3 \cos \theta)^2 = 4 \sin^2 \theta - 12 \sin \theta \cos \theta + 9 \cos^2 \theta \]
Adding these two expansions:
\[ 4 \cos^2 \theta + 9 \sin^2 \theta + 4 \sin^2 \theta + 9 \cos^2 \theta + 12 \cos \theta \sin \theta - 12 \sin \theta \cos \theta \]
This simplifies to:
\[ (4 + 9)(\cos^2 \theta + \sin^2 \theta) = 13 \]
So we have:
\[ \frac{x^2}{25} + \frac{y^2}{16} = 13 \]

### Step 5: Rearranging to Standard Form
Rearranging gives us the standard form of the ellipse:
\[ \frac{x^2}{325} + \frac{y^2}{208} = 1 \]

### Step 6: Identify \( a^2 \) and \( b^2 \)
From the standard form, we identify:
- \( a^2 = 325 \)
- \( b^2 = 208 \)

### Step 7: Calculate the Eccentricity
The eccentricity \( e \) of an ellipse is given by:
\[ e^2 = 1 - \frac{b^2}{a^2} \]
Substituting the values:
\[ e^2 = 1 - \frac{208}{325} \]
Calculating this gives:
\[ e^2 = 1 - 0.64 = 0.36 \]

### Final Answer
Thus, the value of \( e^2 \) is:
\[ \boxed{0.36} \]