1. Given the parabola defined by y^2 = -12x, find the coordinates of the focus, the length of the latus rectum and the coordinates of its endpoints. Find also the equation of the dielectric. Sketch the graph.

2. Find the eccentricity of an ellipse when the length of the latus rectum is 2/3 of the length of the major axis.

y^2 = 4px

has directrix x = -p
focus x = p
the l.r. has length 2p.
No electricity is involved.

the l.r. of an ellipse has length 2b^2/a

recall that b^2+c^2 = a^2
and eccentricity is c/a

To find the answers to these questions, we need to understand the properties of parabolas and ellipses, and use their respective equations.

1. Parabola:
The equation of the parabola is given as y^2 = -12x. We can see that the parabola opens towards the left since the coefficient of x is negative.

To find the coordinates of the focus, we need to know the equation in standard form: (x-h)^2 = 4a(y-k). In our case, h = 0, k = 0, and a = -3 (since 4a = -12, so a = -12 / 4 = -3). Thus, the equation is x^2 = -12y.

Comparing this with the standard form, we see that the coordinates of the focus are (h, k + 1/4a), which in our case is (0, 0 - 1/4(-3)) = (0, 3/4).

The latus rectum is the line segment perpendicular to the axis of the parabola, passing through the focus, and whose endpoints lie on the parabola. Its length is given by 4a, which in our case is 4(-3) = -12.

To find the coordinates of the endpoints of the latus rectum, we need to find the points on the parabola that have the same y-coordinate as the focus. Substituting y = 3/4 into the equation x^2 = -12y, we get x^2 = -12(3/4), which simplifies to x^2 = -9. Taking the square root of both sides, we find x = ±√(-9) = ±3i.

Therefore, the endpoints of the latus rectum are (3i, 3/4) and (-3i, 3/4).

As for the equation of the dielectric, it is not clear what is meant by "the equation of the dielectric" in this context. Please provide more information or clarify the question.

2. Ellipse:
To find the eccentricity of an ellipse given the length of the latus rectum (LR) and the length of the major axis (2a), we can use the formula: eccentricity (e) = LR / 2a.

In this case, we are given that the LR is 2/3 of the length of the major axis. So, if we let LR = (2/3)(2a), then the eccentricity is e = (2/3)(2a) / 2a = 4/6 = 2/3.

Therefore, the eccentricity of the ellipse is 2/3.

Note: The sketching of the graph requires a visual representation, which is not possible through text. I would recommend using graphing software or a graphing calculator to plot the parabola and ellipse based on their equations.