The point of contact of `y=x+3sqrt(2)` with `x^(2)+y^(2)=9` is

Options

A. a) `(3/(sqrt(2)),3/(sqrt(2)))`
B. b) `(3/(sqrt(2)),2/(sqrt(2)))`
C. c) `((-3)/(sqrt(2)),3/(sqrt(2)))`
D. d) `(3/(sqrt(2)),(-3)/(sqrt(2)))`

Answer

To find the point of contact of the line \( y = x + 3\sqrt{2} \) with the circle \( x^2 + y^2 = 9 \), we will follow these steps:

### Step 1: Substitute the equation of the line into the equation of the circle.

We have:
1. The equation of the line: \( y = x + 3\sqrt{2} \) (Equation 1)
2. The equation of the circle: \( x^2 + y^2 = 9 \) (Equation 2)

Substituting Equation 1 into Equation 2:
\[ x^2 + (x + 3\sqrt{2})^2 = 9 \]

### Step 2: Expand the equation.

Expanding \( (x + 3\sqrt{2})^2 \):
\[ (x + 3\sqrt{2})^2 = x^2 + 2 \cdot x \cdot 3\sqrt{2} + (3\sqrt{2})^2 = x^2 + 6\sqrt{2}x + 18 \]

Now substituting back into the circle equation:
\[ x^2 + x^2 + 6\sqrt{2}x + 18 = 9 \]
\[ 2x^2 + 6\sqrt{2}x + 18 - 9 = 0 \]
\[ 2x^2 + 6\sqrt{2}x + 9 = 0 \]

### Step 3: Simplify the equation.

Dividing the entire equation by 2:
\[ x^2 + 3\sqrt{2}x + \frac{9}{2} = 0 \]

### Step 4: Use the quadratic formula to find the values of \( x \).

The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 3\sqrt{2} \), and \( c = \frac{9}{2} \).

Calculating the discriminant:
\[ b^2 - 4ac = (3\sqrt{2})^2 - 4 \cdot 1 \cdot \frac{9}{2} = 18 - 18 = 0 \]

Since the discriminant is zero, there is one repeated root:
\[ x = \frac{-3\sqrt{2}}{2} \]

### Step 5: Find the corresponding \( y \) value.

Using Equation 1 to find \( y \):
\[ y = x + 3\sqrt{2} = -\frac{3\sqrt{2}}{2} + 3\sqrt{2} = -\frac{3\sqrt{2}}{2} + \frac{6\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} \]

### Step 6: Write the point of contact.

Thus, the point of contact is:
\[ \left(-\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right) \]

### Final Answer:
The point of contact of the line with the circle is:
\[ \left(-\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right) \]