`P` is a point on either of the two lines `y - sqrt3|x|=2` at a distance 5 units from their point of intersection The coordinates of the foot of the perpendicular from `P` on the bisector of the angle between them are

Options

A. `(0,(4+5sqrt(3))/2)or (0,(4-5sqrt(3))/2)` depending on which the point P is taken
B. `(0,(4+5sqrt(3))/2)`
C. `(0,(4-5sqrt(3))/2)`
D. `(5/2,(5sqrt(3))/2)`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript while ensuring clarity and completeness in each step.

### Step 1: Identify the equations of the lines
The given lines are:
1. \( y - \sqrt{3}|x| = 2 \) for \( x < 0 \)
2. \( y - \sqrt{3}x = 2 \) for \( x \geq 0 \)

### Step 2: Find the point of intersection
To find the point of intersection, we set the two equations equal to each other.

For \( x = 0 \):
- From the first line: \( y - \sqrt{3} \cdot 0 = 2 \) → \( y = 2 \)
- From the second line: \( y - \sqrt{3} \cdot 0 = 2 \) → \( y = 2 \)

Thus, the point of intersection is \( (0, 2) \).

### Step 3: Determine the distance from point P to the intersection point
Let \( P(x_1, y_1) \) be a point on either line at a distance of 5 units from the intersection point \( (0, 2) \).

Using the distance formula:
\[ \sqrt{(x_1 - 0)^2 + (y_1 - 2)^2} = 5 \]
Squaring both sides gives:
\[ (x_1)^2 + (y_1 - 2)^2 = 25 \]

### Step 4: Substitute for \( y_1 \)
Since \( P \) lies on either of the lines, we can express \( y_1 \) in terms of \( x_1 \):
- For \( x_1 < 0 \): \( y_1 = 2 + \sqrt{3}|x_1| = 2 - \sqrt{3}x_1 \)
- For \( x_1 \geq 0 \): \( y_1 = 2 + \sqrt{3}x_1 \)

Substituting \( y_1 \) into the distance equation:
1. For \( x_1 < 0 \):
\[ (x_1)^2 + (2 - \sqrt{3}x_1 - 2)^2 = 25 \]
Simplifying gives:
\[ (x_1)^2 + (\sqrt{3}x_1)^2 = 25 \implies (1 + 3)(x_1)^2 = 25 \implies 4(x_1)^2 = 25 \implies x_1^2 = \frac{25}{4} \implies x_1 = -\frac{5}{2} \]
Then, substituting back to find \( y_1 \):
\[ y_1 = 2 - \sqrt{3}\left(-\frac{5}{2}\right) = 2 + \frac{5\sqrt{3}}{2} = \frac{4}{2} + \frac{5\sqrt{3}}{2} = \frac{4 + 5\sqrt{3}}{2} \]

2. For \( x_1 \geq 0 \):
\[ (x_1)^2 + (2 + \sqrt{3}x_1 - 2)^2 = 25 \]
This leads to the same calculations, yielding:
\[ x_1 = \frac{5}{2} \quad \text{and} \quad y_1 = 2 + \frac{5\sqrt{3}}{2} \]

### Step 5: Find the angle bisector
The angle bisector of the two lines can be found by equating the two line equations:
\[ y - \sqrt{3}x - 2 = y + \sqrt{3}x - 2 \]
This simplifies to \( x = 0 \), confirming that the angle bisector is the y-axis.

### Step 6: Find the foot of the perpendicular from point P to the angle bisector
Since the angle bisector is the y-axis (i.e., \( x = 0 \)), the foot of the perpendicular from point \( P(x_1, y_1) \) to the y-axis will simply have the x-coordinate of 0, and the y-coordinate will be the same as that of point \( P \).

Thus, the coordinates of the foot of the perpendicular from point \( P \) to the angle bisector are:
\[ (0, y_1) = \left(0, \frac{4 + 5\sqrt{3}}{2}\right) \]

### Final Answer
The coordinates of the foot of the perpendicular from point \( P \) on the bisector of the angle between the lines are:
\[ (0, 4 + \frac{5\sqrt{3}}{2}) \]