To find the equation of the parabola whose latus rectum is the line segment joining the points (-3, 1) and (1, 1), we can follow these steps:
### Step 1: Identify the Points of the Latus Rectum
The points given are L(-3, 1) and L' (1, 1).
### Step 2: Find the Midpoint (Focus) of the Latus Rectum
The focus of the parabola can be found by calculating the midpoint of the line segment joining L and L'. The formula for the midpoint M(x, y) between two points (x1, y1) and (x2, y2) is:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substituting the coordinates of L and L':
\[
M = \left( \frac{-3 + 1}{2}, \frac{1 + 1}{2} \right) = \left( \frac{-2}{2}, \frac{2}{2} \right) = (-1, 1)
\]
So, the focus F of the parabola is at (-1, 1).
### Step 3: Calculate the Length of the Latus Rectum
The length of the latus rectum is the distance between the points L and L'. We can use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates of L and L':
\[
d = \sqrt{(1 - (-3))^2 + (1 - 1)^2} = \sqrt{(1 + 3)^2 + 0} = \sqrt{4^2} = 4
\]
### Step 4: Relate Length of the Latus Rectum to 'a'
The length of the latus rectum (LL') is equal to \(4a\) for a parabola. Since we found the length to be 4, we have:
\[
4a = 4 \implies a = 1
\]
### Step 5: Determine the Vertex of the Parabola
The vertex of the parabola is located at a distance 'a' from the focus along the axis of symmetry. Since the focus is at (-1, 1) and the parabola opens upwards, we can find the vertex V by moving downwards from the focus by 1 unit (since a = 1):
\[
V = (-1, 1 - 1) = (-1, 0)
\]
### Step 6: Write the Equation of the Parabola
The standard form of the equation of a parabola that opens upwards is given by:
\[
(x - h)^2 = 4a(y - k)
\]
where (h, k) is the vertex. Substituting the vertex (-1, 0) and \(a = 1\):
\[
(x + 1)^2 = 4 \cdot 1 \cdot (y - 0)
\]
This simplifies to:
\[
(x + 1)^2 = 4y
\]
### Final Equation
Thus, the equation of the parabola is:
\[
(x + 1)^2 = 4y
\]
