To find the value of \( k \) for which the point \( P(2, k) \) on the ellipse \( x^2 + 2y^2 = 6 \) is nearest to the line \( x + y = 7 \), we can follow these steps:
### Step 1: Identify the ellipse equation
The given ellipse equation is:
\[
x^2 + 2y^2 = 6
\]
This can be rewritten in standard form as:
\[
\frac{x^2}{6} + \frac{y^2}{3} = 1
\]
### Step 2: Parameterize the ellipse
We can parameterize the points on the ellipse using:
\[
x = \sqrt{6} \cos \theta, \quad y = \sqrt{3} \sin \theta
\]
### Step 3: Find the distance from point \( P(2, k) \) to the line \( x + y = 7 \)
The distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[
d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
\]
For the line \( x + y - 7 = 0 \), we have \( A = 1, B = 1, C = -7 \). Thus, the distance from point \( P(2, k) \) is:
\[
d = \frac{|1 \cdot 2 + 1 \cdot k - 7|}{\sqrt{1^2 + 1^2}} = \frac{|2 + k - 7|}{\sqrt{2}} = \frac{|k - 5|}{\sqrt{2}}
\]
### Step 4: Minimize the distance
To minimize the distance \( d \), we need to minimize \( |k - 5| \). The minimum occurs when \( k = 5 \).
### Step 5: Check if the point \( P(2, 5) \) lies on the ellipse
Now we need to check if the point \( P(2, 5) \) lies on the ellipse:
\[
x^2 + 2y^2 = 6
\]
Substituting \( x = 2 \) and \( y = 5 \):
\[
2^2 + 2(5^2) = 4 + 2(25) = 4 + 50 = 54 \quad (\text{not on the ellipse})
\]
### Step 6: Find the correct \( k \) value
Since \( P(2, k) \) must lie on the ellipse, we substitute \( x = 2 \) into the ellipse equation:
\[
2^2 + 2k^2 = 6 \implies 4 + 2k^2 = 6 \implies 2k^2 = 2 \implies k^2 = 1 \implies k = \pm 1
\]
### Step 7: Determine the nearest point
Now we have two potential points on the ellipse: \( P(2, 1) \) and \( P(2, -1) \). We need to find which of these points is closer to the line \( x + y = 7 \).
1. For \( P(2, 1) \):
\[
d = \frac{|2 + 1 - 7|}{\sqrt{2}} = \frac{|3 - 7|}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}
\]
2. For \( P(2, -1) \):
\[
d = \frac{|2 - 1 - 7|}{\sqrt{2}} = \frac{|1 - 7|}{\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2}
\]
### Conclusion
The point \( P(2, 1) \) is closer to the line \( x + y = 7 \) than \( P(2, -1) \). Thus, the value of \( k \) for which the point \( P(2, k) \) on the ellipse is nearest to the line is:
\[
\boxed{1}
\]
