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 the point P(3, 3) on the circle defined by the equation \(x^2 + y^2 = 18\) and two chords PQ and PR, each of length 2 units.

### Step-by-step Solution:

1. **Verify the Point P(3, 3) on the Circle:**
The equation of the circle is \(x^2 + y^2 = 18\).
Substitute \(x = 3\) and \(y = 3\):
\[ 3^2 + 3^2 = 9 + 9 = 18 \]
Since this is true, point P(3, 3) lies on the circle.

**Hint:** Always verify if the given point lies on the circle by substituting its coordinates into the circle's equation.

2. **Identify the Radius of the Circle:**
The radius \(r\) of the circle can be calculated from the equation:
\[ r = \sqrt{18} = 3\sqrt{2} \]

**Hint:** The radius can be found by taking the square root of the right side of the circle's equation.

3. **Draw the Chords PQ and PR:**
We know that the lengths of both chords PQ and PR are 2 units. Let O be the center of the circle (0, 0). The lengths of OP, OQ, and OR (the radius) are all equal to \(3\sqrt{2}\).

4. **Apply the Cosine Rule in Triangle OQP:**
In triangle OQP, we have:
- OP = \(3\sqrt{2}\) (radius)
- OQ = \(3\sqrt{2}\) (radius)
- PQ = 2 (length of the chord)

Using the cosine rule:
\[ PQ^2 = OP^2 + OQ^2 - 2 \cdot OP \cdot OQ \cdot \cos(QOP) \]
Substituting the known values:
\[ 2^2 = (3\sqrt{2})^2 + (3\sqrt{2})^2 - 2 \cdot (3\sqrt{2}) \cdot (3\sqrt{2}) \cdot \cos(QOP) \]
Simplifying:
\[ 4 = 18 + 18 - 18 \cdot \cos(QOP) \]
\[ 4 = 36 - 18 \cdot \cos(QOP) \]
Rearranging gives:
\[ 18 \cdot \cos(QOP) = 36 - 4 = 32 \]
\[ \cos(QOP) = \frac{32}{18} = \frac{16}{9} \]

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

5. **Find the Angle QPR:**
Since QPR is an angle formed by the two chords PQ and PR, and both chords are equal in length, we can use the property of the cosine of angles:
\[ \cos(QPR) = \cos(2 \cdot QOP) \]
Using the double angle formula:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
Substituting \(\cos(QOP)\):
\[ \cos(QPR) = 2 \left(\frac{16}{9}\right)^2 - 1 \]
\[ = 2 \cdot \frac{256}{81} - 1 = \frac{512}{81} - 1 = \frac{512 - 81}{81} = \frac{431}{81} \]

**Hint:** The double angle formula for cosine is useful when dealing with angles formed by equal chords.

6. **Final Result:**
Therefore, the value of \(\cos(QPR)\) is \(\frac{431}{81}\).