Equation of the parabola whose axis is parallel to Y- axis and which passes through the point (1,0),(0,0)and (-2,4) , is

Options

A. `2x^2+2y=3y`
B. `2x^2-2x=3y`
C. `2x^2+2x=y`
D. `2x^2-2x=y`

Answer

To find the equation of the parabola whose axis is parallel to the Y-axis and passes through the points (1,0), (0,0), and (-2,4), 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 first point (1,0)
Substituting the point (1,0) into the equation:
\[ 0 = a(1)^2 + b(1) + c \]
This simplifies to:
\[ 0 = a + b + c \]
This gives us our first equation:
\[ a + b + c = 0 \quad \text{(1)} \]

### Step 3: Substitute the second point (0,0)
Now, substitute the point (0,0):
\[ 0 = a(0)^2 + b(0) + c \]
This simplifies to:
\[ 0 = c \]
So, we find:
\[ c = 0 \quad \text{(2)} \]

### Step 4: Substitute the third point (-2,4)
Next, substitute the point (-2,4):
\[ 4 = a(-2)^2 + b(-2) + c \]
Substituting \( c = 0 \) from equation (2):
\[ 4 = 4a - 2b \]
This gives us our third equation:
\[ 4 = 4a - 2b \quad \text{(3)} \]

### Step 5: Substitute \( c = 0 \) into equation (1)
Now, substitute \( c = 0 \) into equation (1):
\[ a + b + 0 = 0 \]
This simplifies to:
\[ a + b = 0 \]
From this, we can express \( b \) in terms of \( a \):
\[ b = -a \quad \text{(4)} \]

### Step 6: Substitute \( b = -a \) into equation (3)
Now, substitute \( b = -a \) into equation (3):
\[ 4 = 4a - 2(-a) \]
This simplifies to:
\[ 4 = 4a + 2a \]
\[ 4 = 6a \]
Now, solve for \( a \):
\[ a = \frac{4}{6} = \frac{2}{3} \]

### Step 7: Find \( b \)
Using equation (4):
\[ b = -a = -\frac{2}{3} \]

### Step 8: Write the final equation of the parabola
Now we have \( a \), \( b \), and \( c \):
- \( a = \frac{2}{3} \)
- \( b = -\frac{2}{3} \)
- \( c = 0 \)

Thus, the equation of the parabola is:
\[ y = \frac{2}{3}x^2 - \frac{2}{3}x \]

### Final Answer:
The equation of the parabola is:
\[ y = \frac{2}{3}x^2 - \frac{2}{3}x \]

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