The coordinates of an end-point of the latusrectum of the parabola `(y-1)^(2)=4(x+1),` are

Options

A. (0, -3)
B. (0, -1)
C. (0, 1)
D. (1, 3)

Explanation

To find the coordinates of an endpoint of the latus rectum of the parabola given by the equation \((y - 1)^2 = 4(x + 1)\), we can follow these steps:

### Step 1: Identify the standard form of the parabola
The given equation can be rewritten in a standard form. The standard form of a parabola that opens to the right is \(y^2 = 4ax\).

In our case, we have:
\[ (y - 1)^2 = 4(x + 1) \]

### Step 2: Shift the origin
To convert the equation into the standard form, we can shift the origin. Let:
\[ Y = y - 1 \quad \text{and} \quad X = x + 1 \]
Then, the equation becomes:
\[ Y^2 = 4X \]

### Step 3: Identify the value of \(a\)
From the equation \(Y^2 = 4X\), we can see that this is in the form \(Y^2 = 4aX\). Here, we can identify \(4a = 4\), which gives us:
\[ a = 1 \]

### Step 4: Find the endpoints of the latus rectum
For a parabola in the form \(y^2 = 4ax\), the endpoints of the latus rectum are given by the coordinates:
\[ (a, 2a) \quad \text{and} \quad (a, -2a) \]
Substituting \(a = 1\):
\[ (1, 2) \quad \text{and} \quad (1, -2) \]

### Step 5: Convert back to original coordinates
Now we need to convert these coordinates back to the original coordinates:
1. For the point \((1, 2)\):
- \(x = X - 1 = 1 - 1 = 0\)
- \(y = Y + 1 = 2 + 1 = 3\)
- Thus, the first endpoint is \((0, 3)\).

2. For the point \((1, -2)\):
- \(x = X - 1 = 1 - 1 = 0\)
- \(y = Y + 1 = -2 + 1 = -1\)
- Thus, the second endpoint is \((0, -1)\).

### Final Answer
The coordinates of the endpoints of the latus rectum of the parabola \((y - 1)^2 = 4(x + 1)\) are:
\[ (0, 3) \quad \text{and} \quad (0, -1) \]