The focus of the parabola \(\mathrm{x}^{2}-8 \mathrm{x}+2 \mathrm{y}+7=0\) is \(\ldots \ldots \ldots\) (a) \([4,(9 / 2)]\) (b) \([0,(1 / 2)]\) (c) \([4,(9 / 2)]\) (d) \((4,4)\)

To rewrite the equation in vertex form, complete the square with the x terms: Given equation: \(x^2 - 8x + 2y + 7 = 0\) First, rearrange the terms: \( 2y = -x^2+8x-7\) Now move 2 to the other side \( y = \frac{-1}{2}x^2+4x-\frac{7}{2} \) The equation is now in the form \(y=ax^2+bx+c\). Next, complete the square. Rewriting b as \(b=2h\), the vertex form is: \( y = a(x-h)^2 + k \) For our equation, \(a=-\frac{1}{2}\) and \(b=8\). Then we get \(h = \frac{8}{2} = 4\). Now we determine k. We can find it with this formula: \( k = c - \frac{(2ah)^2}{2a} \) \( k = -\frac{7}{2} - \frac{(-\frac{1}{2})(4)^2}{2(-\frac{1}{2})} \) \( k =- \frac{7}{2} - (-4) =- \frac{7}{2} + 4=\frac{1}{2} \) So, the vertex form is: \(y=-\frac{1}{2}(x-4)^2 + \frac{1}{2}\)

The general vertex form is \((x-h)^2 = 4p(y-k)\). Comparing it to our equation, we can identify 4p: \( -\frac{1}{2}(x-4)^2 = 4p(y-\frac{1}{2}) \). Thus: \( 4p=-\frac{1}{2}\). It follows that: \( p = \frac{-1}{8}\).

Since the parabola opens downwards, the focus point will be above the vertex. We found the vertex coordinates as (4,1/2) and the value of p as -1/8. To find the focus, we add the value of p to the y-coordinate of the vertex: Focus Y-coordinate: \(k + p = \frac{1}{2} + \frac{-1}{8}=\frac{3}{8}\). So, the focus point is (4, 3/8).

Comparing our answer to the given options, none of them match the correct focus point we found, which is (4, 3/8). Hence, the correct answer is not listed among the given options.