If y = mx + c is a tangent to the ellipse `x^(2) + 2y^(2) = 6`, them `c^(2) =`

Options

A. `36//m^(2)`
B. `6 m^(2) - 3`
C. `3 m^(2) + 6`
D. `6 m^(2) + 3`

Answer

To solve the problem, we need to find the value of \( c^2 \) given that the line \( y = mx + c \) is a tangent to the ellipse defined by the equation \( x^2 + 2y^2 = 6 \).

### Step-by-Step Solution:

1. **Rewrite the equation of the ellipse**:
The given ellipse equation is:
\[ x^2 + 2y^2 = 6 \]
We can rewrite this in standard form:
\[ \frac{x^2}{6} + \frac{y^2}{3} = 1 \]
This shows that \( a^2 = 6 \) and \( b^2 = 3 \).

2. **Use the formula for the tangent to the ellipse**:
The equation of the tangent to the ellipse can be expressed as:
\[ y = mx \pm \sqrt{a^2 m^2 + b^2} \]
Here, we need to compare this with the line equation \( y = mx + c \).

3. **Identify the relationship between \( c \) and the ellipse parameters**:
From the comparison, we can see that:
\[ c = \sqrt{a^2 m^2 + b^2} \]

4. **Substitute the values of \( a^2 \) and \( b^2 \)**:
We know \( a^2 = 6 \) and \( b^2 = 3 \). Therefore, we can substitute these values into the equation:
\[ c = \sqrt{6m^2 + 3} \]

5. **Find \( c^2 \)**:
To find \( c^2 \), we square both sides:
\[ c^2 = 6m^2 + 3 \]

### Final Answer:
Thus, the value of \( c^2 \) is:
\[ c^2 = 6m^2 + 3 \]