Equation or Parabola whose axis is parallel to y-axis and passing through the points (1, 2), (4, -1) and (2, 3) is

Options

A. `y^(2)=3x`
B. `y^(2)=6x`
C. `y^(2)=12x`
D. `y^(2)=2x`

Answer

To find the equation of the parabola whose axis is parallel to the y-axis and passes through the points (1, 2), (4, -1), and (2, 3), we can follow these steps:

### Step 1: Write the general form 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 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 form a system of equations.

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

2. For the point (4, -1):
\[ -1 = a(4)^2 + b(4) + c \]
This simplifies to:
\[ -1 = 16a + 4b + c \]
(Equation 2)

3. For the point (2, 3):
\[ 3 = a(2)^2 + b(2) + c \]
This simplifies to:
\[ 3 = 4a + 2b + c \]
(Equation 3)

### Step 3: Write the system of equations
Now we have the following system of equations:
1. \( a + b + c = 2 \) (Equation 1)
2. \( 16a + 4b + c = -1 \) (Equation 2)
3. \( 4a + 2b + c = 3 \) (Equation 3)

### Step 4: Eliminate \(c\)
We can eliminate \(c\) by subtracting Equation 1 from Equations 2 and 3.

From Equation 2:
\[ (16a + 4b + c) - (a + b + c) = -1 - 2 \]
This simplifies to:
\[ 15a + 3b = -3 \]
Dividing through by 3 gives:
\[ 5a + b = -1 \] (Equation 4)

From Equation 3:
\[ (4a + 2b + c) - (a + b + c) = 3 - 2 \]
This simplifies to:
\[ 3a + b = 1 \] (Equation 5)

### Step 5: Solve the two-variable system
Now we have a simpler system with two variables:
1. \( 5a + b = -1 \) (Equation 4)
2. \( 3a + b = 1 \) (Equation 5)

Subtract Equation 5 from Equation 4:
\[ (5a + b) - (3a + b) = -1 - 1 \]
This simplifies to:
\[ 2a = -2 \]
Thus, we find:
\[ a = -1 \]

### Step 6: Substitute \(a\) back to find \(b\)
Now substitute \(a = -1\) back into Equation 5:
\[ 3(-1) + b = 1 \]
This simplifies to:
\[ -3 + b = 1 \]
Thus, we find:
\[ b = 4 \]

### Step 7: Substitute \(a\) and \(b\) back to find \(c\)
Now substitute \(a = -1\) and \(b = 4\) back into Equation 1:
\[ -1 + 4 + c = 2 \]
This simplifies to:
\[ 3 + c = 2 \]
Thus, we find:
\[ c = -1 \]

### Step 8: Write the final equation of the parabola
Now we have \(a\), \(b\), and \(c\):
- \(a = -1\)
- \(b = 4\)
- \(c = -1\)

Therefore, the equation of the parabola is:
\[ y = -x^2 + 4x - 1 \]