Question

If r₁, and r₂ are distances of points on the ellipse $5x^2 + 5y^2 + 6xy - 8 = 0$ which are at maximum and minimum distance from the origin, then $r_1 + r_2$ is equal to

• A. 2
• B. 1
• C. 3
• D. 4

Answer 1

Answer: (C)

3

Answer 2

The equation of the ellipse is $5x^2 + 6xy + 5y^2 = 8$. We can write this in matrix form as $\mathbf{x}^T Q \mathbf{x} = 8$, where $\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}$ and $Q = \begin{pmatrix} 5 & 3 \\ 3 & 5 \end{pmatrix}$. The eigenvalues of $Q$ are found by solving $\det(Q - \lambda I) = 0$, which yields $(5-\lambda)^2 - 9 = 0$, giving $\lambda_1 = 2$ and $\lambda_2 = 8$. In the rotated coordinate system, the ellipse equation is $2x'^2 + 8y'^2 = 8$, or $\frac{x'^2}{4} + \frac{y'^2}{1} = 1$. The semi-axes squared are $a'^2 = 4$ and $b'^2 = 1$. The maximum distance from the origin is $r_1 = \sqrt{a'^2} = \sqrt{4} = 2$, and the minimum distance is $r_2 = \sqrt{b'^2} = \sqrt{1} = 1$. Therefore, $r_1 + r_2 = 2 + 1 = 3$.