If `y=m_1x +c_1 and y=m_2x +c_2, m_1 nem_2` , are two common tangents of circle `x^2 + y^2 = 2` and parabola `y^2 = x,` then the value of 8 m,m is equal to :

Options

A. `3+4sqrt2`
B. `-5+6sqrt2`
C. `-4+3sqrt2`
D. `7+6sqrt2`

Answer

To solve the problem, we need to find the value of \(8 |m_1 m_2|\) where \(m_1\) and \(m_2\) are the slopes of the common tangents to the circle \(x^2 + y^2 = 2\) and the parabola \(y^2 = x\).

### Step-by-Step Solution:

1. **Identify the equations of the curves:**
- The circle is given by \(x^2 + y^2 = 2\).
- The parabola is given by \(y^2 = x\).

2. **Write the equation of the tangent to the circle:**
The equation of the tangent to the circle at slope \(m\) is given by:
\[ y = mx + \sqrt{2(1 + m^2)} \]
This represents a line with slope \(m\) that touches the circle.

3. **Write the equation of the tangent to the parabola:**
The equation of the tangent to the parabola \(y^2 = x\) at a point where the slope is \(m\) can be expressed as:
\[ y = mx - \frac{m^2}{4} \]
This represents a line with slope \(m\) that touches the parabola.

4. **Set the two tangent equations equal to each other:**
For the lines to be common tangents, their \(y\) values must be equal at some point:
\[ mx + \sqrt{2(1 + m^2)} = mx - \frac{m^2}{4} \]

5. **Simplify the equation:**
Rearranging gives:
\[ \sqrt{2(1 + m^2)} + \frac{m^2}{4} = 0 \]
Squaring both sides leads to:
\[ 2(1 + m^2) = \left(\frac{m^2}{4}\right)^2 \]

6. **Rearranging and solving for \(m^2\):**
This leads to a quartic equation in \(m^2\):
\[ 32m^4 + 32m^2 - 1 = 0 \]
Let \(x = m^2\), then we have:
\[ 32x^2 + 32x - 1 = 0 \]

7. **Use the quadratic formula to find \(x\):**
The quadratic formula gives us:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-32 \pm \sqrt{32^2 - 4 \cdot 32 \cdot (-1)}}{2 \cdot 32} \]
Simplifying this:
\[ x = \frac{-32 \pm \sqrt{1024 + 128}}{64} = \frac{-32 \pm \sqrt{1152}}{64} \]
\[ = \frac{-32 \pm 24\sqrt{2}}{64} = \frac{-4 \pm 3\sqrt{2}}{8} \]

8. **Find \(m_1 m_2\):**
The product of the slopes \(m_1 m_2\) is given by:
\[ m_1 m_2 = x = \frac{-4 + 3\sqrt{2}}{8} \]

9. **Calculate \(8 |m_1 m_2|\):**
Finally, we calculate:
\[ 8 |m_1 m_2| = 8 \left| \frac{-4 + 3\sqrt{2}}{8} \right| = | -4 + 3\sqrt{2} | \]

### Final Result:
Thus, the value of \(8 |m_1 m_2|\) is:
\[ \boxed{-4 + 3\sqrt{2}} \]