The Cartesian equation of the directrix of the parabola whose parametrix equations are `x=2t+1, y=t^(2)+2`, is

Options

A. y = 2
B. y = 1
C. y = -1
D. y = -2

Answer

To find the Cartesian equation of the directrix of the parabola given by the parametric equations \( x = 2t + 1 \) and \( y = t^2 + 2 \), we can follow these steps:

### Step 1: Express \( t \) in terms of \( x \)
From the equation \( x = 2t + 1 \), we can solve for \( t \):
\[ t = \frac{x - 1}{2} \]

**Hint:** Rearranging the equation to isolate \( t \) helps us substitute it into the equation for \( y \).

### Step 2: Substitute \( t \) into the equation for \( y \)
Now substitute \( t \) into the equation \( y = t^2 + 2 \):
\[ y = \left(\frac{x - 1}{2}\right)^2 + 2 \]

**Hint:** Substitute the expression for \( t \) into the equation for \( y \) to eliminate the parameter.

### Step 3: Simplify the equation for \( y \)
Now simplify the equation:
\[ y = \frac{(x - 1)^2}{4} + 2 \]
Multiply through by 4 to eliminate the fraction:
\[ 4y = (x - 1)^2 + 8 \]

**Hint:** Multiplying through by 4 makes it easier to manipulate the equation.

### Step 4: Rearrange the equation
Rearranging gives:
\[ (x - 1)^2 = 4y - 8 \]

**Hint:** Rearranging helps us identify the standard form of a parabola.

### Step 5: Identify the standard form of the parabola
We can rewrite the equation as:
\[ (x - 1)^2 = 4(y - 2) \]
This is in the form \( (x - h)^2 = 4a(y - k) \), where \( (h, k) \) is the vertex of the parabola.

**Hint:** Recognizing the vertex form of a parabola helps us identify its properties.

### Step 6: Determine the value of \( a \)
From the equation \( (x - 1)^2 = 4(y - 2) \), we can see that \( 4a = 4 \), so \( a = 1 \).

**Hint:** The value of \( a \) is crucial for finding the directrix.

### Step 7: Find the equation of the directrix
For a parabola in the form \( (x - h)^2 = 4a(y - k) \), the directrix is given by the equation:
\[ y = k - a \]
Substituting \( k = 2 \) and \( a = 1 \):
\[ y = 2 - 1 = 1 \]

**Hint:** The directrix is always located at \( k - a \) for parabolas that open upwards.

### Final Answer
The Cartesian equation of the directrix of the parabola is:
\[ \boxed{y = 1} \]