Let F = (0,9) be the focus and the line y = 1 be the directrix. Plot several points P that are three times as far from the focus as they are from the directrix, including the vertices on the y-axis. The configuration of all such P is a hyperbola of eccentricity 3. Use the distance formula to write an equation for the hyperbola. Find the values of a, b, and c for this curve, then calculate the ratio c/a. Is the result what you expected?
since e = 3, c/a = 3
Now, review the properties of an hyperbola.
Given the center at (0,0), with vertex at (0,a), focus at (0,c) and directrix at y=d, we have
(c-a)/(a-d) = 3
c/a = 3
c-d = 8
Solve that for a,c,d, and you have
a=3, c=9, d=1
Since we know the directrix is at y=1, the center must be at y = 1
b^2 = c^2-a^2 = 72
So, our hyperbola is
y^2/9 - x^2/72 = 1
See its properties at
https://www.wolframalpha.com/input/?i=hyperbola+y%5E2%2F9+-+x%5E2%2F72+%3D+1
It does not list the directrices, but we know that
d = a^2/c = 9/9 = 1
To plot several points P that are three times as far from the focus F as they are from the directrix, we can follow these steps:
1. Find the distance between the focus F and the directrix (distance = 1 unit).
2. Multiply the distance by 3 to get the distance from the focus to the plotted points (distance = 3 units).
3. Plot points on the plane that are equidistant from the focus F and the directrix, using the calculated distance. These points will lie on the arms of the hyperbola.
Since the given directrix is a horizontal line y = 1, the hyperbola will have a vertical transverse axis.
Now let's find the equation of the hyperbola by using the properties of the focus and the directrix.
Let's consider a point (x, y) on the hyperbola. The distances from this point to the focus F at (0, 9) and the directrix y = 1 can be calculated using the distance formula:
Distance from the focus: √[(x - 0)^2 + (y - 9)^2]
Distance from the directrix: |y - 1|
According to the given condition, the point (x, y) is three times as far from the focus as it is from the directrix. Putting this in equation form, we get:
√[(x - 0)^2 + (y - 9)^2] = 3 * |y - 1|
To simplify the equation, we can square both sides:
[(x - 0)^2 + (y - 9)^2] = 9 * (y - 1)^2
Expanding and collecting like terms:
x^2 + y^2 - 18y + 81 = 9y^2 - 18y + 9
Rearranging terms and simplifying:
8y^2 - 1 = x^2
This is the equation of the hyperbola.
Now let's find the values of a, b, and c for this curve.
The equation of the hyperbola in standard form is:
(y - k)^2 / a^2 - (x - h)^2 / b^2 = 1
Comparing this to our derived equation, we can see that h = 0 and k = 0.
Therefore, a = √8 and b = 1.
The value of c for a hyperbola can be found using the equation c^2 = a^2 + b^2.
Substituting the values of a and b:
c^2 = (√8)^2 + 1^2
c^2 = 8 + 1
c^2 = 9
Taking the square root of both sides, we find:
c = 3
Finally, let's calculate the ratio c/a:
ratio c/a = 3 / √8
Simplifying this ratio, we get:
ratio c/a = (3 * √8) / 8
Whether the result matches our expectations or not depends on what we initially expected. But based on the derived equation, the value of the ratio c/a is (√8)/8, which is the same as the obtained result.
To plot several points that satisfy the given condition, we need to understand the properties of a hyperbola. Let's start by understanding what it means for a point to be three times as far from the focus as it is from the directrix.
In a hyperbola, the distance from any point P on the hyperbola to the focus (F) is related to the distance from the point to the directrix (D). We can define the distance from P to F as d₁ and the distance from P to D as d₂.
According to the given condition, d₁ = 3d₂. This means that any point P on the hyperbola is three times closer to the focus than it is to the directrix.
Since the focus is at (0,9), we can consider a point P(x, y) on the hyperbola. The distance from P to the focus (d₁) can be calculated using the distance formula:
d₁ = √((x - 0)² + (y - 9)²) = √(x² + (y - 9)²)
The distance from P to the directrix (d₂) can be found by calculating the vertical distance between P and the line y = 1, which is simply:
d₂ = |y - 1| = y - 1
Now, we can equate these distances using the given condition:
√(x² + (y - 9)²) = 3(y - 1)
To simplify further, let's square both sides of the equation:
x² + (y - 9)² = 9(y - 1)²
Expanding and simplifying the equation, we get:
x² + y² - 18y + 81 = 9y² - 18y + 9
Combining like terms:
x² - 9y² + y² + 81 - 9 = 0
x² - 8y² + 72 = 0
This equation represents the hyperbola. By comparing it to the standard form of a hyperbola equation, we can determine the values of 'a', 'b', and 'c' for this curve.
The standard form of a hyperbola equation in the xy-plane is:
(x² / a²) - (y² / b²) = 1
Comparing this to our equation, we can conclude that a² = 8, b² = 72, and c² = a² + b².
Now, let's calculate the ratio c/a:
c/a = √(c²) / √(a²) = √(a² + b²) / a = √(8 + 72) / √8 = √80 / 2√2 = √(16 * 5) / (2√2) = 4√5 / 2√2 = 2√5 / √2
Simplifying by rationalizing the denominator:
(2√5 / √2) * (√2 / √2) = 2√(5 * 2) / 2 = √(10)
The ratio c/a for this hyperbola is √10.
This result aligns with our expectation since the eccentricity (e) of a hyperbola is defined as e = c / a. In this case, c/a = √10, indicating an eccentricity of √10, which matches the given eccentricity of 3.