1 (A). Transforming the general equation of the parabola into standard form:

The given equation is: $y^2 + 4x - 14y = -53$

We will first rearrange the terms and complete the square to convert this into the standard form of a parabola.

1. Rearrange the terms by isolating $x$: $y^2 - 14y = -4x - 53$

2. Complete the square on the left side for the $y$-terms. To complete the square, take half of the coefficient of $y$ (which is $-14$), square it, and add it to both sides. Half of $-14$ is $-7$, and $(-7)^2 = 49$.

   So, we add 49 to both sides: $y^2 - 14y + 49 = -4x - 53 + 49$ Simplifying: $(y - 7)^2 = -4x - 4$ $(y - 7)^2 = -4(x + 1)$

Thus, the equation in standard form is: $(y - 7)^2 = -4(x + 1)$

This is the standard form of a parabola that opens left or right, $(y - k)^2 = 4p(x - h)$, where $(h, k)$ is the vertex and $p$ determines the direction and distance of the focus from the vertex.

1 (B). Finding the coordinates:

From the equation $(y - 7)^2 = -4(x + 1)$, we can extract the following details:

1. Vertex: The vertex $(h, k)$ is given by the point $(-1, 7)$, directly from the form of the equation.

   Vertex: $(-1, 7)$

2. Focus: The general equation is $(y - k)^2 = 4p(x - h)$. In our equation, $4p = -4$, so $p = -1$. Since the parabola opens to the left (because $p$ is negative), the focus is located 1 unit to the left of the vertex. Thus, the focus is at:

   Focus: $(-2, 7)$

3. Directrix: The directrix is a vertical line, located $p$ units to the right of the vertex. Since $p = -1$, the directrix is 1 unit to the right of the vertex, at:

   Directrix: $x = 0$

4. Axis of Symmetry: The axis of symmetry is the horizontal line passing through the vertex, parallel to the $y$-axis. The equation of this line is:

   Axis of Symmetry: $y = 7$

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Would you like further clarification on any of these steps?

5 Related Questions:

1. How do you complete the square for quadratic terms?
2. What does the value of $p$ represent in the standard form of a parabola?
3. How does the equation change if the parabola opens upwards or downwards?
4. What is the general form of the equation for a parabola that opens vertically?
5. How do you derive the equation of the directrix from the standard form of a parabola?

Tip:

Always remember that in the standard form of a parabola $(y - k)^2 = 4p(x - h)$, the sign of $p$ tells you the direction in which the parabola opens.