The equations of the parabolas the extremities of whose latus rectum are `(3, 5)` and `(3,-3)`

Answer

To find the equation of the parabola whose latus rectum has extremities at the points (3, 5) and (3, -3), we can follow these steps:

### Step 1: Identify the coordinates of the extremities of the latus rectum
The extremities of the latus rectum are given as (3, 5) and (3, -3).

### Step 2: Determine the orientation of the parabola
Since the x-coordinates of both points are the same (3), the latus rectum is vertical. This indicates that the parabola opens horizontally (either to the right or to the left).

### Step 3: Calculate the length of the latus rectum
The length of the latus rectum can be calculated using the distance formula:
\[ \text{Length of latus rectum} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the points (3, 5) and (3, -3):
\[ = \sqrt{(3 - 3)^2 + (5 - (-3))^2} = \sqrt{0 + (5 + 3)^2} = \sqrt{8^2} = 8 \]

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

### Step 5: Find the vertex of the parabola
The vertex of the parabola lies at the midpoint of the extremities of the latus rectum. The midpoint \(M\) can be calculated as:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{3 + 3}{2}, \frac{5 + (-3)}{2}\right) = \left(3, 1\right) \]
Thus, the vertex \((h, k)\) is \((3, 1)\).

### Step 6: Write the equation of the parabola
Since the parabola opens horizontally and has its vertex at \((h, k) = (3, 1)\), the standard form of the equation is:
\[ (y - k)^2 = 4a(x - h) \]
Substituting \(h = 3\), \(k = 1\), and \(a = 2\):
\[ (y - 1)^2 = 8(x - 3) \]

### Final Answer
The equation of the parabola is:
\[ (y - 1)^2 = 8(x - 3) \]

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