Two equal chords AB and AC of the circle `x^2 +y^2-6x -8y-24 = 0` are drawn from the point `A(sqrt33 +3,0)`. Another chord PQ is drawn intersecting AB and AC at points R and S, respectively given that `AR=SC=7` and RB = AS = 3 . The value of `PR`/`QS` is

Options

A. 1
B. 1.5
C. 2
D. None of these

Explanation

To solve the problem, we will follow these steps:

### Step 1: Rewrite the equation of the circle
The given equation of the circle is:
\[ x^2 + y^2 - 6x - 8y - 24 = 0 \]
We can rewrite this in standard form by completing the square.

### Step 2: Completing the square
1. For \(x\):
\[ x^2 - 6x \rightarrow (x - 3)^2 - 9 \]
2. For \(y\):
\[ y^2 - 8y \rightarrow (y - 4)^2 - 16 \]
Now substituting back into the equation:
\[ (x - 3)^2 - 9 + (y - 4)^2 - 16 - 24 = 0 \]
This simplifies to:
\[ (x - 3)^2 + (y - 4)^2 = 49 \]
Thus, the center of the circle is \( (3, 4) \) and the radius is \( 7 \).

### Step 3: Identify point A
The point \( A \) is given as \( (\sqrt{33} + 3, 0) \). We need to check if this point lies on the circle:
1. Substitute \( x = \sqrt{33} + 3 \) and \( y = 0 \) into the circle's equation:
\[ (\sqrt{33} + 3 - 3)^2 + (0 - 4)^2 = 49 \]
\[ (\sqrt{33})^2 + 16 = 49 \]
\[ 33 + 16 = 49 \]
This confirms that point A lies on the circle.

### Step 4: Set up the segments
Given:
- \( AR = SC = 7 \)
- \( RB = AS = 3 \)

### Step 5: Calculate lengths
From the segments:
- \( AB = AR + RB = 7 + 3 = 10 \)
- \( AC = AS + SC = 3 + 7 = 10 \)

Since \( AB = AC \), both chords are equal.

### Step 6: Use the power of a point theorem
According to the power of a point theorem:
\[ PR \cdot RQ = AR \cdot RB \]
Let \( PR = x \) and \( RQ = y \). Then:
\[ x \cdot y = 7 \cdot 3 = 21 \]

### Step 7: Set up the second chord
For chord \( PQ \):
\[ QS \cdot SP = AS \cdot SC \]
Let \( QS = a \) and \( SP = b \). Then:
\[ a \cdot b = 3 \cdot 7 = 21 \]

### Step 8: Find the ratio
We need to find the ratio:
\[ \frac{PR}{QS} = \frac{x}{a} \]
From the previous steps, we know:
\[ x \cdot y = 21 \quad \text{and} \quad a \cdot b = 21 \]
Thus, \( PR \cdot RQ = QS \cdot SP \).

Since both products equal 21, we can deduce:
\[ \frac{PR}{QS} = 1 \]

### Final Answer
The value of \( \frac{PR}{QS} \) is \( 1 \).

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