if `r_(1)` and `r_(2)` 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

Options

A. `r_(1)+r_(2)=3`
B. `|r_(1)-r_(2)|=1`
C. `|r_(1)-2r_(2)|=0`
D. `r_(1)+2r_(2)=4`

Explanation

To solve the problem of finding the maximum and minimum distances from the origin to the ellipse given by the equation \(5x^2 + 5y^2 + 6xy - 8 = 0\), we can follow these steps:

### Step 1: Rewrite the ellipse equation
We start with the equation of the ellipse:
\[ 5x^2 + 5y^2 + 6xy - 8 = 0 \]
We can express \(x\) and \(y\) in terms of polar coordinates, where \(x = r \cos \theta\) and \(y = r \sin \theta\).

### Step 2: Substitute polar coordinates into the ellipse equation
Substituting \(x\) and \(y\) into the ellipse equation gives:
\[ 5(r \cos \theta)^2 + 5(r \sin \theta)^2 + 6(r \cos \theta)(r \sin \theta) - 8 = 0 \]
This simplifies to:
\[ 5r^2 \cos^2 \theta + 5r^2 \sin^2 \theta + 6r^2 \cos \theta \sin \theta - 8 = 0 \]

### Step 3: Simplify the equation
Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\), we can simplify the equation:
\[ 5r^2 + 6r^2 \cos \theta \sin \theta - 8 = 0 \]
This can be rewritten as:
\[ 5r^2 + 3r^2 \sin(2\theta) - 8 = 0 \]

### Step 4: Rearrange to find \(r^2\)
Rearranging gives:
\[ 5r^2 + 3r^2 \sin(2\theta) = 8 \]
Factoring out \(r^2\):
\[ r^2(5 + 3 \sin(2\theta)) = 8 \]
Thus, we have:
\[ r^2 = \frac{8}{5 + 3 \sin(2\theta)} \]

### Step 5: Find maximum and minimum values of \(r\)
To find the maximum and minimum distances, we need to analyze the expression \(\frac{8}{5 + 3 \sin(2\theta)}\). The maximum value of \(r^2\) occurs when \(5 + 3 \sin(2\theta)\) is minimized, and the minimum value occurs when \(5 + 3 \sin(2\theta)\) is maximized.

- The maximum value of \(\sin(2\theta)\) is \(1\):
\[ 5 + 3 \cdot 1 = 8 \quad \Rightarrow \quad r^2_{\text{min}} = \frac{8}{8} = 1 \quad \Rightarrow \quad r_{\text{min}} = 1 \]

- The minimum value of \(\sin(2\theta)\) is \(-1\):
\[ 5 + 3 \cdot (-1) = 2 \quad \Rightarrow \quad r^2_{\text{max}} = \frac{8}{2} = 4 \quad \Rightarrow \quad r_{\text{max}} = 2 \]

### Step 6: Conclusion
The maximum distance \(r_1\) from the origin is \(2\) and the minimum distance \(r_2\) is \(1\).

Thus, the final answer is:
\[ r_1 = 2, \quad r_2 = 1 \]