Find the equation of the parabola with directrix x + 1 = 0 and y = 1, and the length of the latus rectum is 4. Sketch the graph.
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Why did the parabola go to therapy?
Because it couldn't focus on its directrix!
To find the equation of the parabola, we first need to determine its vertex and focus.
Given that the directrix is x + 1 = 0, we can rewrite it as x = -1. This implies that the vertex of the parabola is at the point (-1, 1).
The distance from the vertex to the directrix is the same as the distance from the vertex to the focus. Since the latus rectum has a length of 4, the distance between the vertex and the focus is 2.
Therefore, the focus is located at the point (-3, 1).
Now that we have the vertex and focus, we can determine the equation of the parabola using the formula: (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance between the vertex and focus.
Plugging in the values from above, we get: (x + 1)^2 = 4(2)(y - 1).
Simplifying, we have: (x + 1)^2 = 8(y - 1).
Expanding, we get: x^2 + 2x + 1 = 8y - 8.
Rearranging the equation, we have: x^2 + 2x + 8y - 9 = 0.
Thus, the equation of the parabola is x^2 + 2x + 8y - 9 = 0.
To sketch the graph, we can plot the vertex at (-1, 1) and the focus at (-3, 1). Using the equation of the parabola, we can find a few additional points and then draw a smooth curve passing through those points.
I hope this helps! Let me know if you have any further questions.
To find the equation of the parabola with the given conditions, we need to determine the coordinates of the focus and the vertex.
1. Directrix: The equation of the directrix is given as x + 1 = 0. In general, the equation of the directrix for a parabola of the form y^2 = 4ax is x = -a.
So, comparing x + 1 = 0 with x = -a, we can see that a = -1.
2. Vertex: The vertex of the parabola lies at the midpoint between the focus and the directrix. Since the directrix is a horizontal line, the vertex will have the form (h, k), where k is the y-coordinate of the directrix. Thus, the vertex is (-1/2, 1).
3. Focus: The focus of the parabola is a point (h, k + 1/4a). Since we already know the vertex as (-1/2, 1), we can substitute these values into the formula to find the focus. Since a = -1, the focus is given by (h, 1 + 1/4(-1)). Simplifying, we get the focus as (h, 1 - 1/4) or (h, 3/4).
4. Latus Rectum: The length of the latus rectum is given as 4. For a parabola of the form y^2 = 4ax, the length of the latus rectum is 4a.
So, in this case, 4a = 4, and solving for a, we get a = 1.
Now that we have determined the vertex (-1/2, 1), the focus (h, 3/4), and the value of a (1), we can write the equation of the parabola:
The equation of a parabola in vertex form is given as (y - k)^2 = 4a(x - h), where (h, k) is the vertex.
Plugging in the values, we get (y - 1)^2 = 4(1)(x - (-1/2)).
Simplifying, we have (y - 1)^2 = 4(x + 1/2).
To sketch the graph, plot the vertex (-1/2, 1) and the focus (h, 3/4), and draw the parabola using the given equation.
Note: The directrix is the horizontal line y = 1, and the latus rectum is the line segment perpendicular to the axis of the parabola passing through the focus. So the latus rectum will be a horizontal line segment with length 4, centered at the focus (h, 3/4).