The equation of the ellipse is `(x-1) ^(2) + (y -1) ^(2) = 1/9 ((3x + 4y -5)/(5)) ^(2),` then the length of minor axis of this ellipse is

Options

A. `(3)/(sqrt50)`
B. `(6)/(sqrt50)`
C. `(9)/(20)`
D. `(9)/(10)`

Answer

To find the length of the minor axis of the given ellipse, we start with the equation:

\[ (x-1)^2 + (y-1)^2 = \frac{1}{9} \left( \frac{3x + 4y - 5}{5} \right)^2 \]

### Step 1: Rewrite the equation
First, we can simplify the equation. Let's rewrite it:

\[ (x-1)^2 + (y-1)^2 = \frac{1}{9} \cdot \frac{(3x + 4y - 5)^2}{25} \]

Multiplying both sides by 25 to eliminate the denominator:

\[ 25((x-1)^2 + (y-1)^2) = \frac{1}{9} (3x + 4y - 5)^2 \]

### Step 2: Rearranging the equation
Now, let's rearrange the equation:

\[ 225((x-1)^2 + (y-1)^2) = (3x + 4y - 5)^2 \]

### Step 3: Identify the standard form of the ellipse
The standard form of an ellipse centered at \((h, k)\) is:

\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]

From our equation, we can see that the left side represents a circle-like structure, and the right side represents a quadratic form.

### Step 4: Determine the semi-major and semi-minor axes
To find the lengths of the axes, we need to identify \(a\) and \(b\) from the equation.

From the equation:

\[ \frac{(x-1)^2}{\frac{1}{9}} + \frac{(y-1)^2}{\frac{25}{9}} = 1 \]

This indicates that:

- \(a^2 = \frac{25}{9}\) (for the semi-major axis)
- \(b^2 = \frac{1}{9}\) (for the semi-minor axis)

### Step 5: Calculate the lengths of the axes
Now, we can find \(a\) and \(b\):

\[ a = \sqrt{\frac{25}{9}} = \frac{5}{3} \]
\[ b = \sqrt{\frac{1}{9}} = \frac{1}{3} \]

### Step 6: Length of the minor axis
The length of the minor axis is given by \(2b\):

\[ \text{Length of minor axis} = 2b = 2 \times \frac{1}{3} = \frac{2}{3} \]

Thus, the length of the minor axis of the ellipse is:

\[ \boxed{\frac{2}{3}} \]