For the ellipse `4x^(2)+5y^(2)-16x-30y+60=0`, which of the following is true ?

Options

A. Centre=(2,3)
B. Length of major axes is1
C. Eccentricity=`(1)/(sqrt5)`
D. All of these

Answer

To solve the problem, we need to analyze the given equation of the ellipse and determine its properties, such as the center, lengths of the axes, and eccentricity.

### Step-by-Step Solution:

1. **Rewrite the given equation**:
The equation of the ellipse is given as:
\[ 4x^2 + 5y^2 - 16x - 30y + 60 = 0 \]

2. **Rearranging the equation**:
Move the constant term to the right side:
\[ 4x^2 + 5y^2 - 16x - 30y = -60 \]

3. **Completing the square for x**:
For the \(x\) terms:
\[ 4(x^2 - 4x) = 4((x - 2)^2 - 4) = 4(x - 2)^2 - 16 \]
Substitute this back into the equation:
\[ 4(x - 2)^2 - 16 + 5y^2 - 30y = -60 \]
Simplifying gives:
\[ 4(x - 2)^2 + 5y^2 - 30y = -44 \]

4. **Completing the square for y**:
For the \(y\) terms:
\[ 5(y^2 - 6y) = 5((y - 3)^2 - 9) = 5(y - 3)^2 - 45 \]
Substitute this back into the equation:
\[ 4(x - 2)^2 + 5(y - 3)^2 - 45 = -44 \]
Simplifying gives:
\[ 4(x - 2)^2 + 5(y - 3)^2 = 1 \]

5. **Dividing by 1**:
The equation can be rewritten in standard form:
\[ \frac{(x - 2)^2}{\frac{1}{4}} + \frac{(y - 3)^2}{\frac{1}{5}} = 1 \]

6. **Identifying the center**:
From the standard form, we can see that the center of the ellipse is at:
\[ (h, k) = (2, 3) \]

7. **Identifying the lengths of the axes**:
The semi-major axis \(a\) and semi-minor axis \(b\) can be identified:
\[ a = \frac{1}{2}, \quad b = \frac{1}{\sqrt{5}} \]
Therefore, the lengths of the axes are:
- Length of the major axis = \(2b = 2 \times \frac{1}{\sqrt{5}} = \frac{2}{\sqrt{5}}\)
- Length of the minor axis = \(2a = 2 \times \frac{1}{2} = 1\)

8. **Calculating eccentricity**:
The eccentricity \(e\) is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{\left(\frac{1}{\sqrt{5}}\right)^2}{\left(\frac{1}{2}\right)^2}} = \sqrt{1 - \frac{\frac{1}{5}}{\frac{1}{4}}} = \sqrt{1 - \frac{4}{5}} = \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} \]

### Summary of Results:
- Center: \((2, 3)\)
- Length of major axis: \(\frac{2}{\sqrt{5}}\)
- Length of minor axis: \(1\)
- Eccentricity: \(\frac{1}{\sqrt{5}}\)

### Conclusion:
All the options provided in the question regarding the properties of the ellipse are correct.