(A) Transforming the general equation into standard form

We are given the equation of a parabola:

$y^2 + 4x - 14y = -53$

Step 1: Rearrange terms to complete the square

We aim to transform the quadratic in $y$ into a perfect square. First, move all $y$-related terms to one side:

$y^2 - 14y = -4x - 53$

Step 2: Complete the square

To complete the square, take half of the coefficient of $y$, square it, and add that value to both sides. The coefficient of $y$ is -14, so:

$\left(\frac{-14}{2}\right)^2 = 49$

Add 49 to both sides:

$y^2 - 14y + 49 = -4x - 53 + 49$

Simplify:

$(y - 7)^2 = -4x - 4$

Step 3: Simplify and rewrite the equation

Now, factor the right side:

$(y - 7)^2 = -4(x + 1)$

This is the standard form of a parabola equation:

$(y - k)^2 = 4p(x - h)$

where the vertex is $(h, k)$ and $4p$ determines the focal length.

(B) Finding the Coordinates

From the equation:

$(y - 7)^2 = -4(x + 1)$

we can extract the following information:

• The equation is in the form $(y - k)^2 = 4p(x - h)$, where:
  • Vertex: $(h, k) = (-1, 7)$
  • $4p = -4$, so $p = -1$

a. Vertex

The vertex is at $(-1, 7)$.

b. Focus

The focus is located $p$ units away from the vertex along the axis of symmetry. Since $p = -1$, the focus is 1 unit to the left of the vertex (because it's a horizontal parabola opening to the left).

$\text{Focus} = (-1 + (-1), 7) = (-2, 7)$

c. Directrix

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

$x = -1 - (-1) = 0$

So the directrix is the