The envelope of the family of tangents to the curve `y^2=x` is

Options

A. `x + y^ 2=0`
B. `x^2=y`
C. `x^2 + y=0`
D. `y^2 - x=0`

Answer

To find the envelope of the family of tangents to the curve \( y^2 = x \), we can follow these steps:

### Step 1: Understand the curve
The given curve is \( y^2 = x \). This is a standard parabola that opens to the right.

### Step 2: Find the equation of the tangent
To find the equation of the tangent to the curve \( y^2 = x \), we can use the point-slope form of the tangent line. The slope of the tangent line at any point \( (x_0, y_0) \) on the curve can be derived from implicit differentiation.

Differentiating \( y^2 = x \) with respect to \( x \):
\[ 2y \frac{dy}{dx} = 1 \implies \frac{dy}{dx} = \frac{1}{2y} \]
At a point \( (x_0, y_0) \), where \( y_0^2 = x_0 \), the slope of the tangent is \( \frac{1}{2y_0} \).

Using the point-slope form, the equation of the tangent line at point \( (x_0, y_0) \) is:
\[ y - y_0 = \frac{1}{2y_0}(x - x_0) \]

### Step 3: Rearranging the tangent equation
Rearranging the tangent equation gives us:
\[ y = \frac{1}{2y_0}x + y_0 - \frac{x_0}{2y_0} \]
Substituting \( x_0 = y_0^2 \) (since \( y_0^2 = x_0 \)):
\[ y = \frac{1}{2y_0}x + y_0 - \frac{y_0^2}{2y_0} = \frac{1}{2y_0}x + \frac{y_0}{2} \]
Thus, the equation of the tangent can be expressed as:
\[ y = \frac{1}{2y_0}x + \frac{y_0}{2} \]

### Step 4: Parameterize the family of tangents
Let \( c = y_0 \). Then the equation of the tangent becomes:
\[ y = \frac{1}{2c}x + \frac{c}{2} \]

### Step 5: Find the envelope
To find the envelope, we need to eliminate the parameter \( c \). The equation of the tangent can be rewritten as:
\[ 2cy - x - c^2 = 0 \]
Now, we differentiate this equation with respect to \( c \):
\[ \frac{\partial}{\partial c}(2cy - x - c^2) = 2y - 2c = 0 \implies y = c \]

### Step 6: Substitute back to eliminate \( c \)
Substituting \( c = y \) into the tangent equation:
\[ 2y^2 - x - y^2 = 0 \implies y^2 - x = 0 \]

### Conclusion
Thus, the envelope of the family of tangents to the curve \( y^2 = x \) is:
\[ y^2 - x = 0 \]