A hyperbola has centre 'C" and one focus at `P(6,8)`. If its two directrixes are `3x +4y +10 = 0` and `3x +4y - 10 = 0` then `CP =`

Options

A. 14
B. 8
C. 10
D. 6

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the Directrix Equations
The two directrix equations given are:
1. \( D_1: 3x + 4y + 10 = 0 \)
2. \( D_2: 3x + 4y - 10 = 0 \)

### Step 2: Find the Distance Between the Directrices
The distance \( d \) between the two directrices can be calculated using the formula:
\[ d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \]
where \( c_1 \) and \( c_2 \) are the constant terms from the directrix equations, and \( a \) and \( b \) are the coefficients of \( x \) and \( y \) respectively.

Here, \( c_1 = 10 \) and \( c_2 = -10 \), and \( a = 3 \), \( b = 4 \):
\[ d = \frac{|10 - (-10)|}{\sqrt{3^2 + 4^2}} = \frac{20}{\sqrt{9 + 16}} = \frac{20}{\sqrt{25}} = \frac{20}{5} = 4 \]

### Step 3: Relate Distance to Hyperbola Parameters
For a hyperbola, the distance between the directrices is given by \( 2a \). Therefore:
\[ 2a = 4 \implies a = 2 \]

### Step 4: Use the Focus and Directrix Relationship
The distance from the focus \( P(6, 8) \) to the directrix \( D_2: 3x + 4y - 10 = 0 \) is given by:
\[ \text{Distance} = \frac{|3(6) + 4(8) - 10|}{\sqrt{3^2 + 4^2}} = \frac{|18 + 32 - 10|}{\sqrt{9 + 16}} = \frac{|40|}{5} = 8 \]

### Step 5: Relate the Distance to \( a \) and \( e \)
For hyperbolas, the distance from the focus to the directrix can be expressed as:
\[ \text{Distance} = ae \]
Thus, we have:
\[ ae = 8 \]

### Step 6: Find \( e \) Using \( a \)
Since we have \( a = 2 \):
\[ 2e = 8 \implies e = 4 \]

### Step 7: Calculate \( CP \)
The distance \( CP \) (distance from the center to the focus) is given by:
\[ CP = ae \]
Substituting the values we found:
\[ CP = 2 \times 4 = 8 \]

### Final Answer
Thus, the distance \( CP \) is \( 8 \).

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