The vertex of the parabola `y^(2) + 8x - 2y + 17 = 0 ` is (i) (1,-2) (ii) (-2,1) (iii) (1,2) (iv) (2,-1)

Options

A. (1,-2)
B. (-2,1)
C. (1,2)
D. (2,-1)

Answer

To find the vertex of the parabola given by the equation \( y^2 + 8x - 2y + 17 = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
Start with the given equation:
\[ y^2 + 8x - 2y + 17 = 0 \]
Rearranging it gives:
\[ y^2 - 2y + 8x + 17 = 0 \]

### Step 2: Complete the square for the \(y\) terms
We need to complete the square for the \(y\) terms:
\[ y^2 - 2y \]
To complete the square, take half of the coefficient of \(y\) (which is \(-2\)), square it, and add and subtract it:
\[ y^2 - 2y + 1 - 1 = (y - 1)^2 - 1 \]
Now substitute this back into the equation:
\[ (y - 1)^2 - 1 + 8x + 17 = 0 \]
This simplifies to:
\[ (y - 1)^2 + 8x + 16 = 0 \]

### Step 3: Isolate \(x\)
Now isolate \(x\):
\[ 8x = -(y - 1)^2 - 16 \]
Dividing everything by 8 gives:
\[ x = -\frac{1}{8}(y - 1)^2 - 2 \]

### Step 4: Identify the vertex
The equation is now in the vertex form:
\[ x = a(y - k)^2 + h \]
where \(a = -\frac{1}{8}\), \(k = 1\), and \(h = -2\). Thus, the vertex of the parabola is:
\[ (h, k) = (-2, 1) \]

### Conclusion
The vertex of the parabola is \((-2, 1)\). Therefore, the correct answer is:
**(ii) (-2, 1)**.

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