STATEMENT -1 : The farthest point on the circle `x^(2) + y^(2) - 2x - 4y + 4 ` from (0, 0) is (1, 3).
and
STATEMENT-2 : The farthest and nearest points on a circle from a given point are the end points of the diameter through the point.

Options

A. Statement-1 is true, statement-2 is true, Statement -2 is a correct explanation for Statement -1
B. Statement -1 is true, Statement-2 is true , Statement-2 is NOT a correct explanation for statement-1
C. Statement-1 is true, Statement-2 is False
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we need to analyze the two statements given about the circle defined by the equation \(x^2 + y^2 - 2x - 4y + 4 = 0\).

### Step 1: Rewrite the Circle Equation
The first step is to rewrite the equation of the circle in standard form. We will complete the square for both \(x\) and \(y\).

1. Start with the equation:
\[ x^2 - 2x + y^2 - 4y + 4 = 0 \]

2. Rearrange the terms:
\[ (x^2 - 2x) + (y^2 - 4y) = -4 \]

3. Complete the square for \(x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]

4. Complete the square for \(y\):
\[ y^2 - 4y = (y - 2)^2 - 4 \]

5. Substitute back into the equation:
\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 = -4 \]
\[ (x - 1)^2 + (y - 2)^2 - 5 = -4 \]
\[ (x - 1)^2 + (y - 2)^2 = 1 \]

### Step 2: Identify the Center and Radius
From the standard form \((x - 1)^2 + (y - 2)^2 = 1\), we can identify:

- Center of the circle: \((1, 2)\)
- Radius of the circle: \(1\)

### Step 3: Determine the Farthest Point from the Origin
To find the farthest point on the circle from the origin \((0, 0)\), we can use the center and radius:

1. The distance from the center \((1, 2)\) to the origin \((0, 0)\) is:
\[ d = \sqrt{(1 - 0)^2 + (2 - 0)^2} = \sqrt{1 + 4} = \sqrt{5} \]

2. The farthest point on the circle from the origin will be along the line connecting the origin and the center, extending outwards by the radius:
- The direction vector from the origin to the center is \((1, 2)\).
- Normalize this vector:
\[ \text{Unit vector} = \left(\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right) \]

3. The farthest point is then:
\[ \text{Farthest point} = \text{Center} + \text{Radius} \times \text{Unit vector} \]
\[ = (1, 2) + 1 \times \left(\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right) = \left(1 + \frac{1}{\sqrt{5}}, 2 + \frac{2}{\sqrt{5}}\right) \]

### Step 4: Evaluate the Statements
1. **Statement 1** claims that the farthest point is \((1, 3)\). We can see that this point does not match our calculated farthest point. Therefore, **Statement 1 is false**.

2. **Statement 2** states that the farthest and nearest points on a circle from a given point are the endpoints of the diameter through that point. This is a true statement based on the properties of circles. Therefore, **Statement 2 is true**.

### Conclusion
- **Statement 1**: False
- **Statement 2**: True