Given Information:
- Latus rectum length: 1
- Focus: (2,3)
- Directrix: x - y - 3 = 0

Explanation:
To determine the conic, we need to analyze the given information and understand its properties.

Definition of Latus Rectum:
The latus rectum of a conic is the line segment passing through the focus and perpendicular to the major axis, with its endpoints on the conic.

Definition of Focus:
The focus of a conic is a fixed point on the conic that determines its shape.

Definition of Directrix:
The directrix of a conic is a fixed line outside the conic that is equidistant to all points on the conic.

Finding the Shape of the Conic:
1. The latus rectum length is 1, which means the distance between the endpoints of the latus rectum is 1.
2. The focus is given as (2,3).
3. The corresponding directrix is x - y - 3 = 0.

Distance Formula:
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)

Finding the Endpoints of the Latus Rectum:
Let the endpoints of the latus rectum be (x1, y1) and (x2, y2).
Using the distance formula, we have:
√((x2 - x1)^2 + (y2 - y1)^2) = 1

Since the latus rectum is perpendicular to the major axis, the line passing through the focus and the midpoint of the latus rectum is perpendicular to the directrix.

Finding the Midpoint of the Latus Rectum:
The midpoint of the latus rectum is the average of the coordinates of its endpoints.
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Using the midpoint and the equation of the directrix, we can find the equation of the line perpendicular to the directrix passing through the midpoint.

Equation of a Line:
The equation of a line passing through a point (x1, y1) and with a slope m is given by:
y - y1 = m(x - x1)

Equation of the Line Perpendicular to the Directrix:
Using the midpoint of the latus rectum as (x1, y1) and m as the negative reciprocal of the slope of the directrix line, we can find the equation of the line perpendicular to the directrix.

Finding the Intersection Points:
Solving the equations of the line perpendicular to the directrix and the directrix itself will give us the intersection points. These points will be the endpoints of the latus rectum.

Conic Type:
Based on the properties of the conic, we can determine its type:
- If the conic is a parabola, the distance between the vertex and the focus is equal to the distance between the vertex and the directrix.
- If the conic is