Find the equation of tangent to the ellipse `(x^(2))/( 4) + (y^(2))/( 2) = 1 ` that perpendicular to the line `y = x + 1 `. Also, find the point of contact .

Answer

To solve the problem, we need to find the equation of the tangent to the ellipse \(\frac{x^2}{4} + \frac{y^2}{2} = 1\) that is perpendicular to the line \(y = x + 1\). We will also find the point of contact.

### Step 1: Identify the parameters of the ellipse
The given equation of the ellipse is \(\frac{x^2}{4} + \frac{y^2}{2} = 1\).

From this, we can identify:
- \(a^2 = 4\) (thus \(a = 2\))
- \(b^2 = 2\) (thus \(b = \sqrt{2}\))

### Step 2: Determine the slope of the given line
The equation of the line is \(y = x + 1\).

From this, we see that the slope \(m_1\) of the line is \(1\).

### Step 3: Find the slope of the tangent line
Since the tangent line is perpendicular to the given line, the slope \(m\) of the tangent line can be found using the relationship:
\[ m \cdot m_1 = -1 \]
Substituting \(m_1 = 1\):
\[ m \cdot 1 = -1 \implies m = -1 \]

### Step 4: Write the equation of the tangent line
The standard equation of the tangent to the ellipse at a point \((x_0, y_0)\) is given by:
\[ y = mx + \sqrt{a^2 m^2 + b^2} \]
Substituting \(m = -1\), \(a^2 = 4\), and \(b^2 = 2\):
\[ y = -x + \sqrt{4(-1)^2 + 2} = -x + \sqrt{4 + 2} = -x + \sqrt{6} \]
Thus, the equation of the tangent line is:
\[ y = -x + \sqrt{6} \]

### Step 5: Find the point of contact
The coordinates of the point of contact \((x_0, y_0)\) can be found using the formulas:
\[ x_0 = \frac{a^2 m}{\sqrt{a^2 m^2 + b^2}} \quad \text{and} \quad y_0 = \frac{b^2}{\sqrt{a^2 m^2 + b^2}} \]
Substituting \(a^2 = 4\), \(b^2 = 2\), and \(m = -1\):
\[ x_0 = \frac{4 \cdot (-1)}{\sqrt{4(-1)^2 + 2}} = \frac{-4}{\sqrt{6}} \quad \text{and} \quad y_0 = \frac{2}{\sqrt{6}} \]
Thus, the point of contact is:
\[ \left(-\frac{4}{\sqrt{6}}, \frac{2}{\sqrt{6}}\right) \]

### Final Answers
1. The equation of the tangent is:
\[ y = -x + \sqrt{6} \]
2. The point of contact is:
\[ \left(-\frac{4}{\sqrt{6}}, \frac{2}{\sqrt{6}}\right) \]