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 point does not lie in the interior of a circle centered at the origin (0, 0) that passes through the point (13/2, 0).

### Step-by-Step Solution:

1. **Determine the Radius of the Circle**:
The radius \( r \) of the circle can be calculated using the distance from the center (origin) to the point (13/2, 0).
\[ r = \text{Distance from origin to } (13/2, 0) = \sqrt{(13/2 - 0)^2 + (0 - 0)^2} = \sqrt{(13/2)^2} = \frac{13}{2} \]

2. **Identify the Points to Check**:
We need to check the distances of the given points from the origin to determine whether they lie inside, on, or outside the circle. The points are:
- Option 1: \( (9/16, 1) \)
- Option 2: \( (2, 7/3) \)
- Option 3: \( (-3, 4) \)
- Option 4: \( (-6, 5/2) \)

3. **Calculate the Distance for Each Point**:
- **Option 1**:
\[ d_1 = \sqrt{(9/16)^2 + 1^2} = \sqrt{\frac{81}{256} + 1} = \sqrt{\frac{81}{256} + \frac{256}{256}} = \sqrt{\frac{337}{256}} = \frac{\sqrt{337}}{16} \]
- **Option 2**:
\[ d_2 = \sqrt{2^2 + (7/3)^2} = \sqrt{4 + \frac{49}{9}} = \sqrt{\frac{36}{9} + \frac{49}{9}} = \sqrt{\frac{85}{9}} = \frac{\sqrt{85}}{3} \]
- **Option 3**:
\[ d_3 = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- **Option 4**:
\[ d_4 = \sqrt{(-6)^2 + (5/2)^2} = \sqrt{36 + \frac{25}{4}} = \sqrt{\frac{144}{4} + \frac{25}{4}} = \sqrt{\frac{169}{4}} = \frac{13}{2} \]

4. **Compare Distances with the Radius**:
- For **Option 1**:
\(\frac{\sqrt{337}}{16} < \frac{13}{2}\) (inside the circle)
- For **Option 2**:
\(\frac{\sqrt{85}}{3} < \frac{13}{2}\) (inside the circle)
- For **Option 3**:
\(5 < \frac{13}{2}\) (inside the circle)
- For **Option 4**:
\(\frac{13}{2} = \frac{13}{2}\) (on the circle)

5. **Conclusion**:
The point that does not lie in the interior of the circle is **Option 4**: \( (-6, 5/2) \) because it lies on the circle.