Find the equation of the parabola described below. Find the two points that define the latus rectum, and graph the equation. Focus at \( (-5,3) \); directrix the line \( y=1 \) The equation of the parabola is (Type an equation. Use integers or fractions for any numbers in the equation.) The two points that define the latus rectum are (Type ordered pairs. Use a comma to separate answers as needed.) Use the graphing tool to graph the parabola.

To find the equation of the parabola with a focus at \((-5, 3)\) and a directrix at the line \(y = 1\), we can use the definition of a parabola: it is the set of all points that are equidistant from the focus and the directrix. 1. **Determine the vertex**: The vertex of the parabola is located halfway between the focus and the directrix. The focus is at \((-5, 3)\) and the directrix is the line \(y = 1\). The y-coordinate of the vertex is the average of the y-coordinates of the focus and the directrix: \[ y_{\text{vertex}} = \frac{3 + 1}{2} = 2 \] The x-coordinate of the vertex is the same as the x-coordinate of the focus, which is \(-5\). Therefore, the vertex is at \((-5, 2)\). 2. **Determine the orientation**: Since the focus is above the directrix, the parabola opens upwards. 3. **Find the distance \(p\)**: The distance \(p\) from the vertex to the focus (or from the vertex to the directrix) is: \[ p = 3 - 2 = 1 \] 4. **Write the equation**: The standard form of a parabola that opens upwards is given by: \[ (x - h)^2 = 4p(y - k) \] where \((h, k)\) is the vertex. Substituting \(h = -5\), \(k = 2\), and \(p = 1\): \[ (x + 5)^2 = 4(1)(y - 2) \] Simplifying this gives: \[ (x + 5)^2 = 4(y - 2) \] Therefore, the equation of the parabola is: \[ (x + 5)^2 = 4(y - 2) \] 5. **Find the latus rectum**: The latus rectum is a line segment perpendicular to the axis of symmetry of the parabola that passes through the focus. The length of the latus rectum is \(4p\). Since \(p = 1\), the length of the latus rectum is \(4\). The focus is at \((-5, 3)\). The endpoints of the latus rectum can be found by moving \(2\) units (which is half the length of the latus rectum) left and right from the focus along the line \(y = 3\): - Left endpoint: \((-5 - 2, 3) = (-7, 3)\) - Right endpoint: \((-5 + 2, 3) = (-3, 3)\) Thus, the two points that define the latus rectum are: \[ (-7, 3), (-3, 3) \] ### Summary: - The equation of the parabola is: \[ (x + 5)^2 = 4(y - 2) \] - The two points that define the latus rectum are: \[ (-7, 3), (-3, 3) \] ### Graphing the Parabola: To graph the parabola, plot the vertex at \((-5, 2)\), the focus at \((-5, 3)\), the directrix \(y = 1\), and the latus rectum points \((-7, 3)\) and \((-3, 3)\). The parabola will open upwards from the vertex. You can use graphing software or tools to visualize this parabola based on the equation and points provided.