Find the equatin to the parabola whose axis is parallel to the y-xis and which passes through the point `(0,4), (1.9)," and "(-2,6)` and determine its latus rectum.

Answer

To find the equation of the parabola whose axis is parallel to the y-axis and which passes through the points (0, 4), (1, 9), and (-2, 6), we can follow these steps:

### Step 1: Write the general form of the parabola
Since the axis is parallel to the y-axis, we can express the equation of the parabola in the form:
\[ y = ax^2 + bx + c \]

### Step 2: Substitute the first point (0, 4)
Substituting the point (0, 4) into the equation:
\[ 4 = a(0)^2 + b(0) + c \]
This simplifies to:
\[ c = 4 \]

### Step 3: Substitute the second point (1, 9)
Now, substitute the point (1, 9):
\[ 9 = a(1)^2 + b(1) + c \]
Substituting \( c = 4 \):
\[ 9 = a + b + 4 \]
This simplifies to:
\[ a + b = 5 \] (Equation 1)

### Step 4: Substitute the third point (-2, 6)
Next, substitute the point (-2, 6):
\[ 6 = a(-2)^2 + b(-2) + c \]
Substituting \( c = 4 \):
\[ 6 = 4a - 2b + 4 \]
This simplifies to:
\[ 4a - 2b = 2 \]
Dividing the entire equation by 2 gives:
\[ 2a - b = 1 \] (Equation 2)

### Step 5: Solve the system of equations
Now we have two equations:
1. \( a + b = 5 \)
2. \( 2a - b = 1 \)

We can add these two equations:
\[ (a + b) + (2a - b) = 5 + 1 \]
This simplifies to:
\[ 3a = 6 \]
Thus, we find:
\[ a = 2 \]

### Step 6: Substitute back to find b
Now substitute \( a = 2 \) back into Equation 1:
\[ 2 + b = 5 \]
This gives:
\[ b = 3 \]

### Step 7: Write the final equation of the parabola
Now we have \( a = 2 \), \( b = 3 \), and \( c = 4 \). Therefore, the equation of the parabola is:
\[ y = 2x^2 + 3x + 4 \]

### Step 8: Determine the latus rectum
The length of the latus rectum of a parabola in the form \( y = ax^2 + bx + c \) is given by:
\[ \text{Length of latus rectum} = \frac{1}{|a|} \]
Here, \( a = 2 \), so:
\[ \text{Length of latus rectum} = \frac{1}{2} \]

### Final Answer
The equation of the parabola is:
\[ y = 2x^2 + 3x + 4 \]
And the length of the latus rectum is:
\[ \frac{1}{2} \]