Determine the equation of the ellipse with center (−9,8), horizontal major axis of 14, and vertical minor axis of 2.
The standard form equation of an ellipse with center (h,k), horizontal major axis 2a, and vertical minor axis 2b is:
((x-h)^2/a^2) + ((y-k)^2/b^2) = 1
Plugging in the given values, we get:
((x+9)^2/7^2) + ((y-8)^2/1^2) = 1
Simplifying this equation gives the final answer:
(x+9)^2/49 + (y-8)^2/1 = 1
A parabola can be drawn given a focus of
(3,−6) and a directrix of y=−4. What can be said about the parabola?
Since the directrix is a horizontal line, the parabola must open vertically. We can find the vertex of the parabola by finding the midpoint between the focus and the directrix.
The distance between the focus and the directrix is the same for any point on the parabola. This distance is equal to the distance between the point and the focus, measured along a perpendicular line to the directrix. This distance is called the "p" value of the parabola.
We can use this information to find the vertex and the "p" value, and then write the equation of the parabola in standard form.
The vertex is at the midpoint between the focus and the directrix, which is ((3,-6) + (3,-4))/2 = (3,-5).
The distance between the focus (3,-6) and the directrix y=-4 is 2 units. Therefore, the "p" value of the parabola is p = 2.
We can now write the equation of the parabola in standard form:
(y - k)^2 = 4p(x - h)
Substituting the values of the vertex and p, we get:
(y + 5)^2 = 8(x - 3)
Therefore, the equation of the parabola with focus (3,-6) and directrix y=-4 is (y + 5)^2 = 8(x - 3).
A parabola can be drawn given a focus of
(3,−6) and a directrix of y=−4. What can be said about the parabola? The parabola has a vertex at ( , ) has a p-value of blank and it blank
The vertex of the parabola is (3, -5). The p-value of the parabola is 2, which is the distance between the focus and the directrix.
Since the directrix is a horizontal line and the parabola opens vertically, we know that the axis of symmetry is a vertical line passing through the vertex. The equation of the axis of symmetry is x = 3.
The parabola opens upwards because the focus is below the vertex and the directrix is above the vertex.
Therefore, we can summarize the properties of the parabola as:
Vertex: (3, -5)
p-value: 2
Axis of symmetry: x = 3
Opens: Upwards.
To determine the equation of an ellipse, we can use the standard form:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
where (h, k) represents the center of the ellipse, and a and b represent the lengths of the semi-major and semi-minor axes, respectively.
Given that the center is (-9, 8) and the horizontal major axis is 14, and the vertical minor axis is 2, we have:
Center: (h, k) = (-9, 8)
Semi-major axis: a = 14 / 2 = 7
Semi-minor axis: b = 2 / 2 = 1
Now, substituting these values into the standard form, we get:
(x - (-9))^2 / (7^2) + (y - 8)^2 / (1^2) = 1
Simplifying this equation gives us the equation of the ellipse:
(x + 9)^2 / 49 + (y - 8)^2 = 1
To determine the equation of an ellipse, we need to use the standard form:
((x-h)^2)/a^2 + ((y-k)^2)/b^2 = 1
where (h, k) represents the center of the ellipse, and a and b are the lengths of the major and minor axes, respectively.
Given that the center is located at (-9, 8), a horizontal major axis of length 14, and a vertical minor axis of length 2, we can substitute these values into the standard form:
((x - (-9))^2)/((14/2)^2) + ((y - 8)^2)/(2^2) = 1
Simplifying further:
((x + 9)^2)/(7^2) + ((y - 8)^2)/4 = 1
Therefore, the equation of the ellipse with a center at (-9, 8), a horizontal major axis of 14, and a vertical minor axis of 2 is:
((x + 9)^2)/49 + ((y - 8)^2)/4 = 1