The equation of the parabola whose vertex is (1, 1) and focus is (1, 2), is :

Question

Accepted Answer

To determine the equation of the parabola with vertex at (1, 1) and focus at (1, 2), we need to understand the properties of a parabola. A parabola is defined as the set of all points equidistant from a point called the focus and a line called the directrix.

### Explanation for Option 1:

1. **Vertex and Focus**: The given vertex is (1, 1) and the focus is (1, 2). Since the x-coordinates of the vertex and focus are the same, the parabola opens vertically. Specifically, because the focus is above the vertex, the parabola opens upwards.

2. **Equation Form**: For a parabola that opens vertically, the standard form of the equation is $(x - h)^2 = 4p(y - k)$, where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus. Here, $h = 1$ and $k = 1$.

3. **Distance $p$**: The distance from the vertex (1, 1) to the focus (1, 2) is 1 unit. Since the parabola opens upwards, $p = 1$.

4. **Equation Derivation**: Substituting the values into the standard form:
   $$
   (x - 1)^2 = 4 \times 1 \times (y - 1)
   $$
   $$
   (x - 1)^2 = 4(y - 1)
   $$
   Expanding and rearranging:
   $$
   x^2 - 2x + 1 = 4y - 4
   $$
   $$
   x^2 - 2x - 4y + 5 = 0
   $$

This matches Option 1: $x^2 - 2x - 4y + 5 = 0$.

### Explanation for Other Options:

- **Option 2**: $x^2 - 2x + 4y - 5 = 0$
  - This equation implies a parabola that opens downwards, which contradicts the given focus being above the vertex.

- **Option 3**: $x^2 + 2x - 4y - 5 = 0$
  - The presence of $+2x$ instead of $-2x$ suggests a horizontal shift that does not align with the vertex at (1, 1).

- **Option 4**: $x^2 + 2x + 4y - 5 = 0$
  - Similar to Option 3, the $+2x$ term and the $+4y$ term suggest a different orientation and position that do not match the given vertex and focus.

In conclusion, Option 1 is the correct equation for the parabola with the specified vertex and focus, as it accurately reflects the parabola's orientation and position.