Saurabh Koranglekar
Last Activity: 5 Years ago
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:
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.