A line passes through point P(4,5) and makes `30^@` with X-axis. Then the coordinates of the point which is at distance of 4 unit on either side of P is : a. `(4-2sqrt(3),7)`,`(4+2sqrt(3),7)` b. `(4+2sqrt(3),7)`,`(4-2sqrt(3),7)` c. `(4+2sqrt(3),7)`,`(4-2sqrt(3),3)` d. `(4-2sqrt(3),7)`,`(4-2sqrt(3),7)`

Options

A. `(4-2sqrt(3),7)`
B. `(4pm2sqrt(3),7)`
C. `(4+2sqrt(3),7),(4-2sqrt(3),3)`
D. `(4-2sqrt(3),7),(4-2sqrt(3),7)`

Answer

To solve the problem, we need to find the coordinates of points that are at a distance of 4 units on either side of the point P(4, 5) along a line that makes an angle of 30 degrees with the x-axis.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Point P: \( (x_1, y_1) = (4, 5) \)
- Angle with the x-axis: \( \theta = 30^\circ \)
- Distance from point P: \( r = 4 \) units

2. **Use the Distance Formula:**
We know that if a point Q lies on the line making an angle \( \theta \) with the x-axis, then the coordinates of Q can be expressed in terms of P and the distance \( r \):
\[ \frac{x - x_1}{\cos \theta} = \frac{y - y_1}{\sin \theta} = \pm r \]

3. **Calculate Cosine and Sine Values:**
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 30^\circ = \frac{1}{2} \)

4. **Set Up the Equations:**
Using the distance formula, we can write two equations:
\[ \frac{x - 4}{\frac{\sqrt{3}}{2}} = \pm 4 \]
\[ \frac{y - 5}{\frac{1}{2}} = \pm 4 \]

5. **Solve for x:**
- For the positive case:
\[ \frac{x - 4}{\frac{\sqrt{3}}{2}} = 4 \implies x - 4 = 4 \cdot \frac{\sqrt{3}}{2} \implies x - 4 = 2\sqrt{3} \implies x = 4 + 2\sqrt{3} \]
- For the negative case:
\[ \frac{x - 4}{\frac{\sqrt{3}}{2}} = -4 \implies x - 4 = -4 \cdot \frac{\sqrt{3}}{2} \implies x - 4 = -2\sqrt{3} \implies x = 4 - 2\sqrt{3} \]

6. **Solve for y:**
- For the positive case:
\[ \frac{y - 5}{\frac{1}{2}} = 4 \implies y - 5 = 4 \cdot \frac{1}{2} \implies y - 5 = 2 \implies y = 7 \]
- For the negative case:
\[ \frac{y - 5}{\frac{1}{2}} = -4 \implies y - 5 = -4 \cdot \frac{1}{2} \implies y - 5 = -2 \implies y = 3 \]

7. **Final Coordinates:**
The coordinates of the two points that are at a distance of 4 units from P(4, 5) are:
- Point 1: \( (4 + 2\sqrt{3}, 7) \)
- Point 2: \( (4 - 2\sqrt{3}, 3) \)

### Conclusion:
The correct answer is option **c**: \( (4 + 2\sqrt{3}, 7) \) and \( (4 - 2\sqrt{3}, 3) \).