From a point P(3, 3) on the circle `x^(2) + y^(2) =18` , two chords PQ and PR each of 2 units length are drawn on this circle. The value of cos `(/_QPR)` is equal to

Options

A. `1/(3sqrt2)`
B. `-8/9`
C. `(sqrt2)/3`
D. `(-4)/9`

Answer

To solve the problem, we need to find the value of cos(QPR) given that point P(3, 3) lies on the circle defined by the equation \(x^2 + y^2 = 18\) and that two chords PQ and PR each have a length of 2 units.

### Step-by-step Solution:

1. **Identify the Circle's Properties**:
The equation of the circle is given by:
\[ x^2 + y^2 = 18 \]
This can be rewritten to identify the center and radius:
\[ x^2 + y^2 = (3\sqrt{2})^2 \]
Thus, the center of the circle is at (0, 0) and the radius \(r = 3\sqrt{2}\).

**Hint**: Recall that the general form of a circle's equation is \(x^2 + y^2 = r^2\), where (0,0) is the center.

2. **Determine the Lengths**:
We know that the lengths of the chords PQ and PR are both 2 units.

3. **Using the Cosine Rule**:
We can apply the cosine rule in triangle OQP (where O is the center of the circle, and Q and P are points on the circle):
\[ \cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab} \]
Here, \(a\) and \(b\) are the lengths of the radii (both equal to \(3\sqrt{2}\)), and \(c\) is the length of the chord PQ (which is 2).

**Hint**: The cosine rule relates the lengths of the sides of a triangle to the cosine of one of its angles.

4. **Substituting Values**:
Substitute the values into the cosine rule:
\[ \cos(\theta) = \frac{(3\sqrt{2})^2 + (3\sqrt{2})^2 - 2^2}{2 \cdot (3\sqrt{2}) \cdot (3\sqrt{2})} \]
Simplifying:
\[ \cos(\theta) = \frac{18 + 18 - 4}{2 \cdot 18} \]
\[ \cos(\theta) = \frac{32}{36} = \frac{8}{9} \]

**Hint**: Ensure that you simplify the fractions correctly.

5. **Finding cos(2θ)**:
We need to find \(cos(2\theta)\). We can use the double angle formula:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
Substitute \(\cos(\theta) = \frac{8}{9}\):
\[ \cos(2\theta) = 2\left(\frac{8}{9}\right)^2 - 1 \]
\[ = 2\left(\frac{64}{81}\right) - 1 \]
\[ = \frac{128}{81} - 1 = \frac{128}{81} - \frac{81}{81} = \frac{47}{81} \]

**Hint**: Remember that when subtracting fractions, you need a common denominator.

6. **Final Answer**:
Thus, the value of \(\cos(QPR)\) is:
\[ \cos(QPR) = \frac{47}{81} \]