A parabola can be drawn given a focus of (−4,−5) and a directrix of y=9. What can be said about the parabola?
AAAaannndd the bot gets it wrong yet again!
we have p=-7, so
(x+4)^2 = -28(y-2)
I apologize for the mistake. You are correct that the equation of the parabola can be written in the form:
$$(x+4)^2 = -28(y-2)$$
where p = -7. This is the standard form equation of a parabola with a vertical axis of symmetry, opening downwards, and with vertex at (-4, 2).
To understand what can be said about the parabola given the focus and directrix, we need to first understand some important properties of parabolas.
A parabola is a U-shaped curve that can be defined by a focus and a directrix. In general, a parabola is the set of all points that are equidistant to the focus and the directrix.
In this case, the given focus is (-4, -5) and the directrix is the horizontal line y=9. This means that every point on the parabola is equidistant to the focus and the directrix.
To find the equation of the parabola, we can use the definition of a parabola and the distance formula.
Let's name an arbitrary point on the parabola as (x, y). According to the definition of a parabola, the distance between the point (x, y) and the focus (-4, -5) should be equal to the distance between the point (x, y) and the directrix y=9.
We can use the distance formula to calculate these distances. The distance formula between two points (x₁, y₁) and (x₂, y₂) is given by:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Applying the distance formula to the distances between (x, y) and the focus (-4, -5) as well as the directrix y=9, we get:
√((x - (-4))² + (y - (-5))²) = √((x - x)² + (y - 9)²)
Simplifying this equation, we have:
√((x + 4)² + (y + 5)²) = √(x² + (y - 9)²)
Squaring both sides of the equation to eliminate the square roots, we get:
(x + 4)² + (y + 5)² = x² + (y - 9)²
Expanding and simplifying this equation, we have:
x² + 8x + 16 + y² + 10y + 25 = x² + y² - 18y + 81
Cancelling out the x² and y² terms, we get:
8x + 10y - 18y = 81 - 16 - 25
Simplifying further:
8x - 8y = 40
Dividing through by 8, we have:
x - y = 5
So the equation of the parabola is x - y = 5.
From this equation, we can say that the parabola has a slope of 1 (since the coefficient of x is 1). It opens to the right because the coefficient of y is negative (-1). The vertex of the parabola is (5, 0), which is the point where the parabola intersects the axis of symmetry, which in this case is the line y=x.
In conclusion, the given parabola with a focus of (-4, -5) and a directrix of y=9 is a right-opening parabola given by the equation x - y = 5. It has a vertex at (5, 0) and its axis of symmetry is the line y=x.
The vertex of the parabola is halfway between the focus and the directrix, so the vertex is at (−4,2). The parabola opens downwards because the directrix is below the focus. The equation of the parabola can be written as:
$$(y-2) = -\frac{1}{4}(x+4)^2$$
or
$$y = -\frac{1}{4}(x+4)^2 + 2$$
This is a standard form equation of a parabola with a horizontal axis of symmetry.