If p be the length of the perpendicular from the origin on the straight line x + 2 by `+2p = 0` , then what is the value of b ?

Options

A. `(1)/(p)`
B. p
C. `(1)/(2)`
D. `(sqrt3)/(2)`

Answer

To find the value of \( b \) given the length of the perpendicular from the origin to the line \( x + 2by + 2p = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The line equation is given as:
\[ x + 2by + 2p = 0 \]
Here, we can identify:
- \( a = 1 \)
- \( b = 2b \)
- \( c = 2p \)

### Step 2: Use the formula for the length of the perpendicular
The formula for the length of the perpendicular \( P \) from a point \( (x_1, y_1) \) to the line \( ax + by + c = 0 \) is given by:
\[ P = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]
Substituting \( (x_1, y_1) = (0, 0) \):
\[ P = \frac{|a(0) + b(0) + c|}{\sqrt{a^2 + b^2}} = \frac{|c|}{\sqrt{a^2 + b^2}} = \frac{|2p|}{\sqrt{1^2 + (2b)^2}} \]

### Step 3: Simplify the expression
This simplifies to:
\[ P = \frac{2p}{\sqrt{1 + 4b^2}} \]

### Step 4: Set the length of the perpendicular equal to \( p \)
According to the problem, the length of the perpendicular is given as \( p \):
\[ p = \frac{2p}{\sqrt{1 + 4b^2}} \]

### Step 5: Cancel \( p \) (assuming \( p \neq 0 \))
Dividing both sides by \( p \):
\[ 1 = \frac{2}{\sqrt{1 + 4b^2}} \]

### Step 6: Cross-multiply and square both sides
Cross-multiplying gives:
\[ \sqrt{1 + 4b^2} = 2 \]
Now, squaring both sides:
\[ 1 + 4b^2 = 4 \]

### Step 7: Solve for \( b^2 \)
Rearranging gives:
\[ 4b^2 = 4 - 1 = 3 \]
Thus:
\[ b^2 = \frac{3}{4} \]

### Step 8: Find \( b \)
Taking the square root of both sides:
\[ b = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Conclusion
The value of \( b \) is:
\[ \boxed{\frac{\sqrt{3}}{2}} \]