The slopes of the tangents drawn from (4, 1) to the ellipse `x^(2)+2y^(2)=6` are

Options

A. `1,1//5`
B. `-1,-1//5`
C. `-1,1//5`
D. `1,-1//5`

Answer

To find the slopes of the tangents drawn from the point (4, 1) to the ellipse given by the equation \( x^2 + 2y^2 = 6 \), we will follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Equation of the Ellipse**:
The given equation of the ellipse is \( x^2 + 2y^2 = 6 \). We can rewrite it in standard form by dividing both sides by 6:
\[ \frac{x^2}{6} + \frac{y^2}{3} = 1 \]
Here, we identify \( a^2 = 6 \) and \( b^2 = 3 \).

**Hint**: Remember that the standard form of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

2. **Equation of the Tangent Line**:
The equation of the tangent to the ellipse at a point with slope \( m \) is given by:
\[ y = mx \pm \sqrt{a^2 m^2 + b^2} \]
Substituting \( a^2 = 6 \) and \( b^2 = 3 \):
\[ y = mx \pm \sqrt{6m^2 + 3} \]

3. **Substituting the Point (4, 1)**:
Since the tangent passes through the point (4, 1), we substitute \( x = 4 \) and \( y = 1 \) into the tangent equation:
\[ 1 = 4m \pm \sqrt{6m^2 + 3} \]

4. **Rearranging the Equation**:
Rearranging gives:
\[ 1 - 4m = \pm \sqrt{6m^2 + 3} \]

5. **Squaring Both Sides**:
Squaring both sides to eliminate the square root gives:
\[ (1 - 4m)^2 = 6m^2 + 3 \]
Expanding the left side:
\[ 1 - 8m + 16m^2 = 6m^2 + 3 \]

6. **Combining Like Terms**:
Rearranging the equation gives:
\[ 16m^2 - 6m^2 - 8m + 1 - 3 = 0 \]
Simplifying this results in:
\[ 10m^2 - 8m - 2 = 0 \]

7. **Solving the Quadratic Equation**:
We can factor or use the quadratic formula to solve for \( m \):
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 10 \cdot (-2)}}{2 \cdot 10} \]
Calculating the discriminant:
\[ = \frac{8 \pm \sqrt{64 + 80}}{20} = \frac{8 \pm \sqrt{144}}{20} = \frac{8 \pm 12}{20} \]
This gives us two solutions:
\[ m_1 = \frac{20}{20} = 1 \quad \text{and} \quad m_2 = \frac{-4}{20} = -\frac{1}{5} \]

8. **Final Answer**:
The slopes of the tangents drawn from the point (4, 1) to the ellipse are:
\[ m = 1 \quad \text{and} \quad m = -\frac{1}{5} \]

### Summary of Slopes:
The slopes of the tangents are \( 1 \) and \( -\frac{1}{5} \).