find the eqyuation of hyperbola traced by points that moves so the difference to (0,0) and (11,11) is 11
just plug in your specification:
√(x^2+y^2) = √((x-11)^2 + (y-11)^2) + 11
square both sides to get
x^2+y^2 = (x-11)^2 + (y-11)^2 + 121 + 22√((x-11)^2 + (y-11)^2)
collect terms to get and divide by 11 to get
2x+2y-33 = 2√((x-11)^2 + (y-11)^2)
square again and collect terms and you wind up with
44x + 44y - 8xy = 121
which is, as expected, a rotated hyperbola.
You could have started out by specifying that the foci were at ±11/√2 and the shifted and rotated it, but that would have been a bit more trouble.
To find the equation of a hyperbola with a center at the origin, we need to use the standard equation:
(x^2 / a^2) - (y^2 / b^2) = 1 for a hyperbola opening horizontally
or
(y^2 / a^2) - (x^2 / b^2) = 1 for a hyperbola opening vertically.
The given condition states that the distance between the points (0,0) and (11,11) is 11. This means that the hyperbola will have its vertices at these points.
To find the values of a and b, we can use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the given points, we can substitute the values into the distance formula to find a and b:
11 = sqrt((11 - 0)^2 + (11 - 0)^2)
Simplifying this equation:
11 = sqrt(121 + 121)
11 = sqrt(242)
Squaring both sides to eliminate the square root:
121 = 242
This is not a valid equation, which means there is an error in the problem or assumption. It is not possible for the difference between the distance of two points from the origin to equal the same value as the distance between the two points.
Please check the problem statement or provide additional information.
To find the equation of the hyperbola traced by points that move such that the difference between their distances to (0, 0) and (11, 11) is 11, we can follow these steps:
Step 1: Understand the properties of a hyperbola
A hyperbola is a conic section with eccentricity greater than 1. Its equation generally takes the form: x^2/a^2 - y^2/b^2 = 1 or y^2/b^2 - x^2/a^2 = 1, depending on the orientation.
Step 2: Determine the center of the hyperbola
Since the given points are (0, 0) and (11, 11), we know that the center of the hyperbola will be the midpoint of these two points: ((0+11)/2, (0+11)/2) = (5.5, 5.5).
Step 3: Calculate the distance between the center and one of the given points
Using the distance formula, we can find the distance between the center (5.5, 5.5) and one of the given points, say (0, 0):
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((5.5 - 0)^2 + (5.5 - 0)^2)
Distance = sqrt(5.5^2 + 5.5^2)
Distance = sqrt(60.5)
Step 4: Determine the distance between the center and the focus
Since the eccentricity of a hyperbola is greater than 1, we can calculate the distance between the center and the focus using the following formula:
Distance between center and focus = sqrt(a^2 + b^2)
Step 5: Find the values of a and b
We can set up an equation using the information given: the difference between the distances to (0, 0) and (11, 11) is 11.
Distance to (0, 0) - Distance to (11, 11) = 11
sqrt(a^2 + b^2) - sqrt(a^2 + b^2 + 121) = 11
Step 6: Solve for a and b
Squaring both sides of the equation above, we get:
a^2 + b^2 - (a^2 + b^2 + 121) = (11 + sqrt(a^2 + b^2))^2
-121 = 22sqrt(a^2 + b^2)
(sqrt(a^2 + b^2))^2 = (121/22)^2
a^2 + b^2 = (121/22)^2
Step 7: Write the equation of the hyperbola
Since the center is (5.5, 5.5), we have the equation of the hyperbola in the form:
(x - h)^2/a^2 - (y - k)^2/b^2 = 1
Plugging in the values, we get:
(x - 5.5)^2/a^2 - (y - 5.5)^2/b^2 = 1
But from Step 6, we know that a^2 + b^2 = (121/22)^2, so we can rewrite the equation as:
(x - 5.5)^2/(121/22)^2 - (y - 5.5)^2/(121/22)^2 = 1
Therefore, the equation of the hyperbola traced by the points that move such that the difference between their distances to (0, 0) and (11, 11) is 11 is:
(x - 5.5)^2/(121/22)^2 - (y - 5.5)^2/(121/22)^2 = 1.