If A represents the area of the ellipse 3x^2+4xy+3y^2=1 then find the value of (3root5÷pie)×A

To find the area of the ellipse represented by the equation 3x² + 4xy + 3y² = 1, we first need to convert this quadratic form into a standard form of an ellipse. The area of an ellipse can then be calculated using the formula A = πab, where a and b are the semi-major and semi-minor axes, respectively.

Step-by-Step Breakdown

Identifying the Coefficients

The given equation is a quadratic form in x and y. We can represent it in a matrix form as follows:

M =
| 3 2 |
| 2 3 |

This arises because the equation can be expressed as a quadratic form: X^T M X = 1, where X = [x, y]^T.

Finding Eigenvalues

To diagonalize this matrix, we need to find its eigenvalues by solving the characteristic polynomial given by det(M - λI) = 0, where I is the identity matrix. Let's compute it:

det( | 3-λ 2 | ) = (3-λ)(3-λ) - 4 = λ² - 6λ + 5 = 0

Solving this gives us:

• λ₁ = 5
• λ₂ = 1

Calculating the Area

The semi-major and semi-minor axes can be derived from the eigenvalues. The area of the ellipse can be calculated using the formula:

A = π * (1/√λ₁) * (1/√λ₂) = π * (1/√5) * (1/1) = π/√5.

Final Calculation

We need to find the value of (3√5/π) × A. Plugging in the area we calculated:

(3√5/π) × (π/√5) = 3.

Final Result

Thus, the value of (3√5/π) × A is equal to 3.