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 equations to isolate the variable can help in substituting into other equations.

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

**Hint:** When substituting, ensure you square the entire expression correctly and add any constants.

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

**Hint:** Multiplying through by a common factor can help simplify the equation.

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

**Hint:** Recognizing the standard form of a parabola helps identify key parameters.

### Step 5: Identify the value of \( a \)
From the standard form \( (x - 1)^2 = 4(y - 2) \), we can see that \( 4a = 4 \), thus:
\[ a = 1 \]

**Hint:** The coefficient of the linear term in the standard form gives you the value of \( a \).

### Step 6: Find the equation of the directrix
For a parabola that opens upwards, the directrix is given by the equation:
\[ y = k - a \]
Here, \( k = 2 \) and \( a = 1 \):
\[ y = 2 - 1 = 1 \]

**Hint:** Remember the relationship between the vertex, focus, and directrix when dealing with parabolas.

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