Find the equation to the parabola whose axis parallel to the y-axis and which passes through the points (0,4)(1,9) and (4,5) and determine its latusrectum.

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 (4, 5), we can follow these steps:

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

### Step 2: Substitute the points into the equation
We will substitute each of the given points into the equation to create a system of equations.

1. For the point (0, 4):
\[ 4 = a(0)^2 + b(0) + c \]
This simplifies to:
\[ c = 4 \]

2. For 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)

3. For the point (4, 5):
\[ 5 = a(4)^2 + b(4) + c \]
Substituting \( c = 4 \):
\[ 5 = 16a + 4b + 4 \]
This simplifies to:
\[ 16a + 4b = 1 \]
Dividing through by 4 gives:
\[ 4a + b = \frac{1}{4} \] (Equation 2)

### Step 3: Solve the system of equations
Now we have a system of two equations:
1. \( a + b = 5 \)
2. \( 4a + b = \frac{1}{4} \)

We can solve this system by elimination or substitution. Let's use elimination.

Subtract Equation 1 from Equation 2:
\[ (4a + b) - (a + b) = \frac{1}{4} - 5 \]
This simplifies to:
\[ 3a = \frac{1}{4} - \frac{20}{4} = -\frac{19}{4} \]
Thus,
\[ a = -\frac{19}{12} \]

Now substitute \( a \) back into Equation 1 to find \( b \):
\[ -\frac{19}{12} + b = 5 \]
\[ b = 5 + \frac{19}{12} = \frac{60}{12} + \frac{19}{12} = \frac{79}{12} \]

### Step 4: Write the final equation of the parabola
Now we have:
- \( a = -\frac{19}{12} \)
- \( b = \frac{79}{12} \)
- \( c = 4 \)

Substituting these values into the equation \( y = ax^2 + bx + c \):
\[ y = -\frac{19}{12}x^2 + \frac{79}{12}x + 4 \]

To express this in a standard form:
\[ 12y = -19x^2 + 79x + 48 \]
Rearranging gives:
\[ 19x^2 - 79x + 12y = 48 \]

### Step 5: Determine the latus rectum
The latus rectum \( L \) of a parabola is given by the formula:
\[ L = \frac{1}{|a|} \]
Here, \( a = -\frac{19}{12} \), so:
\[ L = \frac{1}{\left| -\frac{19}{12} \right|} = \frac{12}{19} \]

### Final Answer
The equation of the parabola is:
\[ 19x^2 - 79x + 12y = 48 \]
And the length of the latus rectum is:
\[ \frac{12}{19} \]