write equation of hyperbola with cebter c(4,2) vertex (4,5) equation of one asymptote 4y-3x=4
AAAaannndd the bot gets it wrong yet again!
c=5, not b.
(y-2)^2/9 - (x-4)^2/16 = 1
To write the equation of a hyperbola, we need the coordinates of the center, the coordinates of a vertex, and the equation of one of the asymptotes.
Given:
Center: (4,2)
Vertex: (4,5)
Equation of one asymptote: 4y - 3x = 4
Step 1: Determine the type of hyperbola
To determine the type of hyperbola, we compare the x-coordinates of the vertices and the y-coordinates of the vertices. Since the x-coordinates of the vertices are the same (both 4 in this case), it indicates a horizontal hyperbola.
Step 2: Find the coordinates of the other vertex
Since the hyperbola is horizontal, we can find the coordinates of the other vertex by subtracting the y-coordinate of the center from the y-coordinate of the given vertex:
Vertex: (4, y)
Center: (4, 2)
Given Vertex: (4, 5)
y - 2 = 5 - 2
y - 2 = 3
y = 5
The coordinates of the other vertex are (4, 5).
Step 3: Find the value of 'a'
The distance between the center and each vertex represents the value of 'a'. In this case, the x-coordinates of both vertices are the same, so the value of 'a' is the difference between the y-coordinate of the center and either vertex.
a = 5 - 2
a = 3
Step 4: Find the value of 'c'
The distance between the center and each focus represents the value of 'c'. In this case, the distance between the center and vertex is 'a' (since there is no slant in this case).
c = 3
Step 5: Write the equation of the hyperbola
The equation of a hyperbola with the center at (h, k), vertices at (h ± a, k), and 'c' represents the distance between the center and either focus is given by:
(x - h)² / a² - (y - k)² / b² = 1
Since the hyperbola is horizontal, 'a' represents the distance between the center and each vertex. Thus, the equation becomes:
(x - 4)² / 3² - (y - 2)² / b² = 1
To find the value of 'b', we can use the relationship between 'a', 'b', and 'c' in a hyperbola:
c² = a² + b²
Squaring both sides:
3² = 3² + b²
9 = 9 + b²
b² = 0
Since b² = 0, it indicates that 'b' is equal to 0.
Therefore, the equation of the hyperbola is:
(x - 4)² / 3² - (y - 2)² / 0 = 1
Simplifying further, we get:
(x - 4)² / 9 = 1
Thus, the equation of the hyperbola is (x - 4)² / 9 = 1.
To write the equation of a hyperbola, we need to know its center, vertices, and one asymptote.
The given hyperbola has a center at C(4,2) and a vertex at (4,5). We can use this information to determine the distance between the center and the vertex, which is the value of 'a' in the equation. In this case, 'a' = 5 - 2 = 3.
The equation of a hyperbola with center (h, k), where 'a' represents the distance from the center to either vertex and 'b' represents the distance from the center to either foci, is given by:
[(x - h)² / a²] - [(y - k)² / b²] = 1 (for a horizontal hyperbola)
[(y - k)² / a²] - [(x - h)² / b²] = 1 (for a vertical hyperbola)
Plugging in the values, we have:
[(x - 4)² / 3²] - [(y - 2)² / b²] = 1
Now, we need to determine the value of 'b' and the equation of the second asymptote. Since one asymptote is given as 4y - 3x = 4, we know that the slope of the asymptotes is ±(b/a).
The slope of the given asymptote is 4/3, which means that ±(b/a) = 4/3.
To find the value of 'b', we can use the formula b = ±(a√m), where 'm' is the slope of the given asymptote.
Plugging in the values, we have:
4/3 = b/3
b = 4
Therefore, the equation of the hyperbola with the given center, vertex, and one asymptote is:
[(x - 4)² / 3²] - [(y - 2)² / 4²] = 1