Vertical Hyperbola Formula
The equation for a vertical hyperbola is given by:
(x – h)^2 / a^2 – (y – k)^2 / b^2 = 1
where (h, k) represents the coordinates of the center of the hyperbola, and a and b represent the distances from the center to the vertices along the x-axis and y-axis respectively
The equation for a vertical hyperbola is given by:
(x – h)^2 / a^2 – (y – k)^2 / b^2 = 1
where (h, k) represents the coordinates of the center of the hyperbola, and a and b represent the distances from the center to the vertices along the x-axis and y-axis respectively.
In this equation, the term (x – h)^2 / a^2 is positive, which means that the x-term is subtracted from the equation. This causes the graph of the hyperbola to open vertically.
Similarly, the term (y – k)^2 / b^2 is negative, which means that the y-term is also subtracted from the equation. This further confirms that the graph will open vertically.
The center of the hyperbola is located at the point (h, k), which represents the midpoint between the two vertices.
The vertices of the hyperbola are located at (h, k ± a), which means that the vertices lie along the line parallel to the y-axis, vertically above and below the center.
The foci of the hyperbola are located at (h, k ± c), where c is determined by the equation c^2 = a^2 + b^2. The foci lie along the same line as the vertices but are closer to the center.
The asymptotes of the hyperbola are given by the equations y = ± (b / a) * (x – h) + k. These lines represent the paths that the graph of the hyperbola approaches but never reaches.
In summary, the formula for a vertical hyperbola is:
(x – h)^2 / a^2 – (y – k)^2 / b^2 = 1
Where:
– (h, k) denotes the center of the hyperbola.
– a represents the distance from the center to the vertices along the x-axis.
– b represents the distance from the center to the vertices along the y-axis.
– The vertices are located at (h, k ± a).
– The foci are located at (h, k ± c) where c = sqrt(a^2 + b^2).
– The asymptotes are represented by the equations y = ± (b / a) * (x – h) + k.
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