The ends of Iatusrectum of parabola are (6, 7) and (6, -1) then equation of parabola is

Options

A. `(y-3)^(2)-8(x-4)`
B. `(y+3)^(2)=8(x-4)`
C. `(y-3)^(2)=-8(x-4)`
D. `(y+3)^(2)=-8(x+4)`

Answer

To find the equation of the parabola given the endpoints of the latus rectum, we can follow these steps:

### Step 1: Identify the endpoints of the latus rectum
The endpoints of the latus rectum are given as (6, 7) and (6, -1).

### Step 2: Find the midpoint (focus)
The focus of the parabola is the midpoint of the latus rectum. We can calculate it using the formula for the midpoint:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of the endpoints:
\[ \text{Focus} = \left( \frac{6 + 6}{2}, \frac{7 + (-1)}{2} \right) = \left( \frac{12}{2}, \frac{6}{2} \right) = (6, 3) \]

### Step 3: Calculate the length of the latus rectum
The length of the latus rectum can be calculated using the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ \text{Length} = \sqrt{(6 - 6)^2 + (7 - (-1))^2} = \sqrt{0 + (7 + 1)^2} = \sqrt{8^2} = 8 \]

### Step 4: Relate the length of the latus rectum to 'a'
The length of the latus rectum is given by \(4a\). Therefore, we have:
\[ 4a = 8 \implies a = 2 \]

### Step 5: Determine the vertex
The vertex of the parabola lies halfway between the focus and the directrix. Since the focus is at (6, 3) and the parabola opens vertically, the vertex can be calculated as:
\[ \text{Vertex} = (h, k) = (6, 3 - a) = (6, 3 - 2) = (6, 1) \]

### Step 6: Write the equation of the parabola
The standard form of the equation of a parabola that opens upwards is:
\[ (y - k)^2 = 4a(x - h) \]
Substituting the values of \(h\), \(k\), and \(a\):
\[ (y - 3)^2 = 4 \cdot 2 (x - 6) \]
This simplifies to:
\[ (y - 3)^2 = 8(x - 6) \]

### Final Equation
Thus, the equation of the parabola is:
\[ (y - 3)^2 = 8(x - 6) \]

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