the differential equation representing all the the tangents to the parabola `y^(2)=2x` is

Options

A. `2x((dy)/(dx))^(2)-2y(dy)/(dx)+1=0`
B. `2x((dy)/(dx))^(2)+2y(dy)/(dx)-1=0`
C. `2x((dy)/(dx))^(2)-2y(dy)/(dx)-1=0`
D. `2x((dy)/(dx))^(2)+2y(dy)/(dx)+1=0`

Answer

To find the differential equation representing all the tangents to the parabola \( y^2 = 2x \), we can follow these steps:

### Step 1: Differentiate the given equation
We start with the equation of the parabola:
\[ y^2 = 2x \]
Now, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(2x) \]
Using the chain rule on the left side, we get:
\[ 2y \frac{dy}{dx} = 2 \]

### Step 2: Solve for \(\frac{dy}{dx}\)
Now, we can simplify the equation:
\[ 2y \frac{dy}{dx} = 2 \]
Dividing both sides by \( 2y \) (assuming \( y \neq 0 \)):
\[ \frac{dy}{dx} = \frac{1}{y} \]

### Step 3: Express the tangent line
The equation of the tangent line to the parabola at a point \( (x_0, y_0) \) can be expressed as:
\[ y - y_0 = m(x - x_0) \]
where \( m \) is the slope of the tangent line. From our previous step, we know that the slope \( m = \frac{1}{y_0} \). The point on the parabola can be expressed in terms of \( y_0 \):
\[ x_0 = \frac{y_0^2}{2} \]
Substituting \( x_0 \) and \( m \) into the tangent line equation, we have:
\[ y - y_0 = \frac{1}{y_0} \left( x - \frac{y_0^2}{2} \right) \]

### Step 4: Rearranging the tangent equation
Rearranging the tangent line equation gives:
\[ y - y_0 = \frac{1}{y_0} x - \frac{y_0}{2} \]
Multiplying through by \( y_0 \) to eliminate the fraction:
\[ y_0 y - y_0^2 = x - \frac{y_0^2}{2} \]
Rearranging gives:
\[ y_0 y - x + \frac{y_0^2}{2} = y_0^2 \]
Thus, we can express this as:
\[ y_0 y - x + \frac{y_0^2}{2} - y_0^2 = 0 \]
This simplifies to:
\[ y_0 y - x - \frac{y_0^2}{2} = 0 \]

### Step 5: Eliminate \( y_0 \)
Now, we need to eliminate \( y_0 \) from this equation. From \( y_0^2 = 2x \), we can express \( y_0 \) in terms of \( x \):
\[ y_0 = \sqrt{2x} \]
Substituting \( y_0 \) back into the tangent equation:
\[ \sqrt{2x} y - x - \frac{2x}{2} = 0 \]
This simplifies to:
\[ \sqrt{2x} y - x - x = 0 \]
or
\[ \sqrt{2x} y - 2x = 0 \]

### Step 6: Final form of the differential equation
Rearranging gives us:
\[ \sqrt{2x} y = 2x \]
Squaring both sides to eliminate the square root:
\[ 2xy^2 = 4x^2 \]
Dividing through by \( x \) (assuming \( x \neq 0 \)):
\[ 2y^2 = 4x \]
Thus, the final differential equation representing all the tangents to the parabola \( y^2 = 2x \) is:
\[ y^2 = 2x \]