Find the equation of the parabola whose axis is parallel to X-axis and which passes through the point (0,4),(1,9) and (-2,6) . Also, find its latusrectum.

Answer

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

### Step 1: General Form of the Parabola
Since the axis of the parabola is parallel to the X-axis, the equation can be expressed in the form:
\[ y = a(x - h)^2 + k \]
where (h, k) is the vertex of the parabola.

### Step 2: Substitute Points into the Equation
We will use the given points to create a system of equations.

1. For the point (0, 4):
\[ 4 = a(0 - h)^2 + k \]
\[ 4 = ah^2 + k \] (Equation 1)

2. For the point (1, 9):
\[ 9 = a(1 - h)^2 + k \]
\[ 9 = a(1 - 2h + h^2) + k \]
\[ 9 = a(1 - 2h + h^2) + k \] (Equation 2)

3. For the point (-2, 6):
\[ 6 = a(-2 - h)^2 + k \]
\[ 6 = a(4 + 4h + h^2) + k \] (Equation 3)

### Step 3: Simplify the Equations
Now we have three equations:
1. \( k = 4 - ah^2 \)
2. \( 9 = a(1 - 2h + h^2) + k \)
3. \( 6 = a(4 + 4h + h^2) + k \)

Substituting \( k \) from Equation 1 into Equations 2 and 3:

From Equation 2:
\[ 9 = a(1 - 2h + h^2) + (4 - ah^2) \]
\[ 9 = a(1 - 2h) + 4 \]
\[ 5 = a(1 - 2h) \]
\[ a = \frac{5}{1 - 2h} \] (Equation 4)

From Equation 3:
\[ 6 = a(4 + 4h + h^2) + (4 - ah^2) \]
\[ 6 = a(4 + 4h) + 4 \]
\[ 2 = a(4 + 4h) \]
\[ a = \frac{2}{4 + 4h} \] (Equation 5)

### Step 4: Equate the Values of 'a'
Now we can equate the two expressions for 'a' from Equations 4 and 5:
\[ \frac{5}{1 - 2h} = \frac{2}{4 + 4h} \]

Cross-multiplying gives:
\[ 5(4 + 4h) = 2(1 - 2h) \]
\[ 20 + 20h = 2 - 4h \]
\[ 24h = -18 \]
\[ h = -\frac{3}{4} \]

### Step 5: Find 'a' and 'k'
Substituting \( h = -\frac{3}{4} \) into Equation 4 to find 'a':
\[ a = \frac{5}{1 - 2(-\frac{3}{4})} = \frac{5}{1 + \frac{3}{2}} = \frac{5}{\frac{5}{2}} = 2 \]

Now substituting \( a \) back into Equation 1 to find \( k \):
\[ k = 4 - 2(-\frac{3}{4})^2 = 4 - 2 \cdot \frac{9}{16} = 4 - \frac{9}{8} = \frac{32}{8} - \frac{9}{8} = \frac{23}{8} \]

### Step 6: Write the Final Equation
Now we have \( a = 2 \), \( h = -\frac{3}{4} \), and \( k = \frac{23}{8} \). The equation of the parabola is:
\[ y = 2\left(x + \frac{3}{4}\right)^2 + \frac{23}{8} \]

### Step 7: Find the Latus Rectum
The length of the latus rectum of a parabola is given by \( \frac{4}{|a|} \):
\[ \text{Latus Rectum} = \frac{4}{2} = 2 \]

### Final Answer
The equation of the parabola is:
\[ y = 2\left(x + \frac{3}{4}\right)^2 + \frac{23}{8} \]
And the length of the latus rectum is:
\[ 2 \]