To solve the given problem and extract the properties of the parabola, we will start by rewriting the equation in a standard form and then derive each required property step by step.

The given equation is: $3y^2 - 18y + 12x + 3 = 0$

Step 1: Rewrite in Standard Form

We begin by isolating the terms involving $x$ and $y$, focusing on making the quadratic term in $y$ appear as part of a perfect square.

$3y^2 - 18y + 12x + 3 = 0$ Move $12x + 3$ to the other side: $3y^2 - 18y = -12x - 3$ Factor out the 3 from the terms on the left-hand side: $3(y^2 - 6y) = -12x - 3$ Complete the square inside the parentheses. To complete the square, take half of the coefficient of $y$, square it, and add and subtract the same value inside the equation. Here, half of $-6$ is $-3$, and $(-3)^2 = 9$.

Add and subtract 9 inside the parentheses: $3[(y^2 - 6y + 9) - 9] = -12x - 3$ Simplify this: $3(y - 3)^2 - 27 = -12x - 3$ Now move $-27$ to the right-hand side: $3(y - 3)^2 = -12x + 24$ Finally, divide through by 3 to get the equation into standard form: $(y - 3)^2 = -4(x - 2)$ This is the equation of a parabola in standard form $(y - k)^2 = 4p(x - h)$, which opens horizontally.

Step 2: Identify Properties

1. Standard Form:
   $(y - 3)^2 = -4(x - 2)$

2. Opening:
   Since the coefficient of $(y - 3)^2$ is negative, the parabola opens left.

3. Vertex:
   The vertex is $(h, k)$, which corresponds to $(2, 3)$.

4. Distance between the vertex and focus (p):
   From the standard form, $4p = -4$, so $p = -1$.
   The distance between the vertex and the focus is 1 unit.

5. Focus:
   The focus lies 1 unit to the left of the vertex, as the parabola opens left.
   So, the focus is at $(2 - 1, 3) = (1, 3)$.

6. Equation of Axis of Symmetry:
   The axis of symmetry is the horizontal line passing through the vertex.
   The equation is $y = 3$.

7. Length of Latus Rectum:
   The length of the latus rectum is $|4p| = 4$ units.

8. Endpoints of Latus Rectum:
   The endpoints of the latus rectum are 2 units above and 2 units below the focus (since the length is 4 units and is symmetric about the focus).
   The endpoints are $(1, 3 + 2) = (1, 5)$ and $(1, 3 - 2) = (1, 1)$.

9. Equation of Directrix:
   The directrix is a vertical line located $p = 1$ unit to the right of the vertex, at $x = 2 + 1 = 3$.
   The equation of the directrix is $x = 3$.

Summary of Properties:

• Standard Form: $(y - 3)^2 = -4(x - 2)$
• Opening: Left
• Vertex: $(2, 3)$
• Distance between Vertex and Focus: 1 unit
• Focus: $(1, 3)$
• Equation of Axis of Symmetry: $y = 3$
• Length of Latus Rectum: 4 units
• Endpoints of Latus Rectum: $(1, 5)$ and $(1, 1)$
• Equation of Directrix: $x = 3$

Would you like any further clarification or details?

Related Questions:

1. How do we find the equation of a parabola if given its focus and directrix?
2. How does the direction of the parabola's opening change with the coefficient of $x$ or $y$?
3. What is the significance of the latus rectum in parabola geometry?
4. How can we determine the focal distance in more complex conic sections like ellipses and hyperbolas?
5. How does completing the square help in deriving the standard form of a conic?

Tip:

When graphing a parabola, always start by identifying the vertex, as it serves as the foundation for determining the focus, directrix, and other key features.