Find the equation of parabola with its axis parallel to x - axis and passing through the points `(-2,1) ,(1,2) and (-1,3)` .

Answer

To find the equation of the parabola with its axis parallel to the x-axis and passing through the points (-2, 1), (1, 2), and (-1, 3), we can follow these steps:

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

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

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

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

3. For the point (-1, 3):
\[ -1 = a(3^2) + b(3) + c \]
\[ -1 = 9a + 3b + c \]
(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. \( 4a + 2b + c = 1 \) (Equation 2)
3. \( 9a + 3b + c = -1 \) (Equation 3)

### Step 4: Eliminate \(c\)
We can eliminate \(c\) from the equations. From Equation 1, we can express \(c\) as:
\[ c = -2 - a - b \]

Now substitute \(c\) into Equations 2 and 3.

Substituting into Equation 2:
\[ 4a + 2b + (-2 - a - b) = 1 \]
\[ 4a + 2b - 2 - a - b = 1 \]
\[ 3a + b - 2 = 1 \]
\[ 3a + b = 3 \] (Equation 4)

Substituting into Equation 3:
\[ 9a + 3b + (-2 - a - b) = -1 \]
\[ 9a + 3b - 2 - a - b = -1 \]
\[ 8a + 2b - 2 = -1 \]
\[ 8a + 2b = 1 \] (Equation 5)

### Step 5: Solve the system of equations
Now we have a simpler system of equations:
1. \( 3a + b = 3 \) (Equation 4)
2. \( 8a + 2b = 1 \) (Equation 5)

From Equation 4, we can express \(b\) in terms of \(a\):
\[ b = 3 - 3a \]

Now substitute \(b\) into Equation 5:
\[ 8a + 2(3 - 3a) = 1 \]
\[ 8a + 6 - 6a = 1 \]
\[ 2a + 6 = 1 \]
\[ 2a = 1 - 6 \]
\[ 2a = -5 \]
\[ a = -\frac{5}{2} \]

### Step 6: Find \(b\) and \(c\)
Now substitute \(a\) back into Equation 4 to find \(b\):
\[ 3(-\frac{5}{2}) + b = 3 \]
\[ -\frac{15}{2} + b = 3 \]
\[ b = 3 + \frac{15}{2} \]
\[ b = \frac{6}{2} + \frac{15}{2} \]
\[ b = \frac{21}{2} \]

Now substitute \(a\) and \(b\) back into Equation 1 to find \(c\):
\[ -\frac{5}{2} + \frac{21}{2} + c = -2 \]
\[ \frac{16}{2} + c = -2 \]
\[ 8 + c = -2 \]
\[ c = -2 - 8 \]
\[ c = -10 \]

### Step 7: Write the final equation
Now we have:
\[ a = -\frac{5}{2}, \quad b = \frac{21}{2}, \quad c = -10 \]
Thus, the equation of the parabola is:
\[ x = -\frac{5}{2}y^2 + \frac{21}{2}y - 10 \]

### Final Answer:
The equation of the parabola is:
\[ x = -\frac{5}{2}y^2 + \frac{21}{2}y - 10 \]