The equation of the tangent to the curve `y ( 1 + x^(2)) = 2 - x ` , where it crosses the x- axis, is

Options

A. x + 5y = 2
B. x - 5y = 2
C. 5x - y = 2
D. 5x + y = 2

Answer

To find the equation of the tangent to the curve \( y(1 + x^2) = 2 - x \) at the point where it crosses the x-axis, we can follow these steps:

### Step 1: Determine the point of intersection with the x-axis
To find where the curve crosses the x-axis, we set \( y = 0 \) in the equation of the curve:

\[ 0(1 + x^2) = 2 - x \]

This simplifies to:

\[ 0 = 2 - x \]

Solving for \( x \):

\[ x = 2 \]

Thus, the point where the curve intersects the x-axis is \( (2, 0) \).

### Step 2: Differentiate the equation of the curve
We start with the equation of the curve:

\[ y(1 + x^2) = 2 - x \]

We will differentiate both sides with respect to \( x \). Using the product rule on the left side:

\[ \frac{d}{dx}[y(1 + x^2)] = \frac{dy}{dx}(1 + x^2) + y(2x) = -1 \]

This gives us:

\[ \frac{dy}{dx}(1 + x^2) + 2xy = -1 \]

### Step 3: Substitute the point \( (2, 0) \) into the derivative
Now we will substitute \( x = 2 \) and \( y = 0 \) into the differentiated equation to find the slope of the tangent:

\[ \frac{dy}{dx}(1 + 2^2) + 2(2)(0) = -1 \]

This simplifies to:

\[ \frac{dy}{dx}(1 + 4) = -1 \]

\[ 5\frac{dy}{dx} = -1 \]

Thus, we find:

\[ \frac{dy}{dx} = -\frac{1}{5} \]

### Step 4: Write the equation of the tangent line
Now that we have the slope of the tangent line and the point \( (2, 0) \), we can use the point-slope form of the equation of a line:

\[ y - y_1 = m(x - x_1) \]

Substituting \( m = -\frac{1}{5} \), \( x_1 = 2 \), and \( y_1 = 0 \):

\[ y - 0 = -\frac{1}{5}(x - 2) \]

This simplifies to:

\[ y = -\frac{1}{5}x + \frac{2}{5} \]

### Step 5: Rearranging to standard form
To express this in standard form, we can multiply through by 5 to eliminate the fraction:

\[ 5y = -x + 2 \]

Rearranging gives:

\[ x + 5y = 2 \]

### Final Answer
Thus, the equation of the tangent to the curve where it crosses the x-axis is:

\[ \boxed{x + 5y = 2} \]