If a circle drawn with origin as the centre passes through `(13/(2),0)`, then the point which does not lie in the interior of the circle is

Options

A. `((-3)/(4),1)`
B. `(2,(7)/(3))`
C. `(5,(-1)/(2))`
D. `(-6,(5)/(2))`

Explanation

To solve the problem, we need to determine which of the given points does not lie in the interior of the circle centered at the origin (0, 0) and passing through the point (13/2, 0).

### Step-by-Step Solution:

1. **Find the Radius of the Circle**:
The radius of the circle is the distance from the center (0, 0) to the point (13/2, 0).
\[ \text{Radius} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{\left(\frac{13}{2} - 0\right)^2 + (0 - 0)^2} = \sqrt{\left(\frac{13}{2}\right)^2} = \frac{13}{2} = 6.5 \]

2. **Calculate the Distances of Each Point from the Origin**:
We will calculate the distance of each given point from the origin (0, 0) and compare it with the radius (6.5).

- **Point A: (-3, 4, 1)**:
\[ \text{Distance} = \sqrt{(-3 - 0)^2 + (4 - 0)^2 + (1 - 0)^2} = \sqrt{(-3)^2 + 4^2 + 1^2} = \sqrt{9 + 16 + 1} = \sqrt{26} \approx 5.1 \]
(This point lies in the interior of the circle since 5.1 < 6.5)

- **Point B: (2, 7/3, 3)**:
\[ \text{Distance} = \sqrt{(2 - 0)^2 + \left(\frac{7}{3} - 0\right)^2 + (3 - 0)^2} = \sqrt{2^2 + \left(\frac{7}{3}\right)^2 + 3^2} = \sqrt{4 + \frac{49}{9} + 9} = \sqrt{4 + 5.444 + 9} = \sqrt{18.444} \approx 4.29 \]
(This point lies in the interior of the circle since 4.29 < 6.5)

- **Point C: (5, -1/2, 2)**:
\[ \text{Distance} = \sqrt{(5 - 0)^2 + \left(-\frac{1}{2} - 0\right)^2 + (2 - 0)^2} = \sqrt{5^2 + \left(-\frac{1}{2}\right)^2 + 2^2} = \sqrt{25 + \frac{1}{4} + 4} = \sqrt{29.25} \approx 5.41 \]
(This point lies in the interior of the circle since 5.41 < 6.5)

- **Point D: (-6, 5/2)**:
\[ \text{Distance} = \sqrt{(-6 - 0)^2 + \left(\frac{5}{2} - 0\right)^2} = \sqrt{(-6)^2 + \left(\frac{5}{2}\right)^2} = \sqrt{36 + \frac{25}{4}} = \sqrt{36 + 6.25} = \sqrt{42.25} \approx 6.5 \]
(This point lies on the boundary of the circle since 6.5 = 6.5)

3. **Conclusion**:
The only point that does not lie in the interior of the circle is Point D (-6, 5/2) because it lies exactly on the boundary of the circle.

### Final Answer:
The point which does not lie in the interior of the circle is **Point D: (-6, 5/2)**.