Observed the following statements
I: The intercepts of the circle `x^(2)+y^(2)-4x+13=0` on y-axis is `sqrt(7)`
II: The intercept made by the circle `x^(2)+y^(2)-4x-8y+13=0` on x-axis is 15
III: The straight line `y=x+1` cuts the circle `x^(2)+y^(2)=1` in two, two distinct points, then truness, falseness of the above statements are

Options

A. F,T,F
B. F,F,T
C. T,T,F
D. T,T,T

Answer

To determine the truthfulness of the given statements, we will analyze each statement step by step.

### Statement I: The intercepts of the circle \(x^2 + y^2 - 4x + 13 = 0\) on the y-axis is \(\sqrt{7}\).

1. **Rewrite the equation**: Start with the equation of the circle:
\[ x^2 + y^2 - 4x + 13 = 0 \]
Rearranging gives:
\[ (x^2 - 4x) + y^2 + 13 = 0 \]

2. **Complete the square for x**:
\[ (x^2 - 4x + 4) + y^2 + 13 - 4 = 0 \]
This simplifies to:
\[ (x - 2)^2 + y^2 + 9 = 0 \]
or
\[ (x - 2)^2 + y^2 = -9 \]
Since the right side is negative, this circle does not exist in the real plane.

3. **Find y-intercepts**: To find the y-intercepts, set \(x = 0\):
\[ (0 - 2)^2 + y^2 = -9 \implies 4 + y^2 = -9 \implies y^2 = -13 \]
This gives imaginary values for \(y\).

**Conclusion for Statement I**: False.

### Statement II: The intercept made by the circle \(x^2 + y^2 - 4x - 8y + 13 = 0\) on the x-axis is 15.

1. **Rewrite the equation**: Start with the equation of the circle:
\[ x^2 + y^2 - 4x - 8y + 13 = 0 \]
Rearranging gives:
\[ (x^2 - 4x) + (y^2 - 8y) + 13 = 0 \]

2. **Complete the square for x and y**:
\[ (x^2 - 4x + 4) + (y^2 - 8y + 16) + 13 - 4 - 16 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 4)^2 = 7 \]
This represents a circle centered at (2, 4) with a radius of \(\sqrt{7}\).

3. **Find x-intercepts**: To find the x-intercepts, set \(y = 0\):
\[ (x - 2)^2 + (0 - 4)^2 = 7 \implies (x - 2)^2 + 16 = 7 \implies (x - 2)^2 = -9 \]
This again gives imaginary values for \(x\).

**Conclusion for Statement II**: False.

### Statement III: The straight line \(y = x + 1\) cuts the circle \(x^2 + y^2 = 1\) in two distinct points.

1. **Substitute the line equation into the circle equation**:
\[ x^2 + (x + 1)^2 = 1 \]
Expanding gives:
\[ x^2 + (x^2 + 2x + 1) = 1 \implies 2x^2 + 2x + 1 - 1 = 0 \implies 2x^2 + 2x = 0 \]

2. **Factor the equation**:
\[ 2x(x + 1) = 0 \]
This gives solutions:
\[ x = 0 \quad \text{or} \quad x = -1 \]

3. **Find corresponding y-values**:
- For \(x = 0\), \(y = 0 + 1 = 1\) → Point (0, 1).
- For \(x = -1\), \(y = -1 + 1 = 0\) → Point (-1, 0).

**Conclusion for Statement III**: True.

### Final Summary:
- Statement I: False
- Statement II: False
- Statement III: True

Thus, the correct option is that both the first and second statements are false, and the third statement is true.