If a hyperbola is translated horizontally or vertically from the center, the parameters $a$a, $b$b, and $c$c still have the same meaning. However, we must take into account that the center of the hyperbola has moved.
Graph of a hyperbola centred at $\left(h,k\right)$(h,k) |
Given the following definitions for $h$h and $k$k,
and remembering the values,
the table below summarizes the characteristics of a translated hyperbola in both orientations:
Orientation | Horizontal Major Axis | Vertical Major Axis |
---|---|---|
Standard form | $\frac{\left(x-h\right)^2}{a^2}-\frac{\left(y-k\right)^2}{b^2}=1$(x−h)2a2−(y−k)2b2=1 | $\frac{\left(y-k\right)^2}{a^2}-\frac{\left(x-h\right)^2}{b^2}=1$(y−k)2a2−(x−h)2b2=1 |
Center | $\left(h,k\right)$(h,k) | $\left(h,k\right)$(h,k) |
Foci | $\left(h+c,k\right)$(h+c,k) and $\left(h-c,k\right)$(h−c,k) | $\left(h,k+c\right)$(h,k+c) and $\left(h,k-c\right)$(h,k−c) |
Vertices | $\left(h+a,k\right)$(h+a,k) and $\left(k-a,k\right)$(k−a,k) | $\left(h,k+a\right)$(h,k+a) and $\left(h,k-a\right)$(h,k−a) |
Transverse axis | $y=k$y=k | $x=h$x=h |
Conjugate axis | $x=h$x=h | $y=k$y=k |
Asymptotes | $y-k=\pm\frac{b}{a}(x-h)$y−k=±ba(x−h) | $y-k=\pm\frac{a}{b}(x-h)$y−k=±ab(x−h) |
Essentially, the information is the same as the central hyperbola. But the values of $h$h and $k$k are added to the $x$x and $y$y-values (respectively) for each characteristic.
Once we establish certain information about the hyperbola, we can use the relationships summarized in the tables to determine the graph of the hyperbola.
The hyperbola $\frac{x^2}{25}-\frac{y^2}{100}=1$x225−y2100=1 when translated $3$3 units to the right and $4$4 units up will be described by the equation $\frac{\left(x-3\right)^2}{25}-\frac{\left(y-4\right)^2}{100}=1$(x−3)225−(y−4)2100=1. Find the centre, vertices, foci and asymptotes of the translated hyperbola. Then draw the graph of the hyperbola.
Hyperbola centred at the origin and translated hyperbola with its vertices and centre marked |
Graph of a hyperbola $\frac{\left(x-3\right)^2}{25}-\frac{\left(y-4\right)^2}{100}=1$(x−3)225−(y−4)2100=1 |
The graph of the hyperbola $\frac{\left(y-k\right)^2}{a^2}-\frac{\left(x-h\right)^2}{b^2}=1$(y−k)2a2−(x−h)2b2=1 is the same graph as the graph of $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$y2a2−x2b2=1, except the center is at $\left(h,k\right)$(h,k) rather than at the origin.
Given the graph of $\frac{y^2}{16}-\frac{x^2}{4}=1$y216−x24=1, find the graph of $\frac{\left(y-5\right)^2}{16}-\frac{x^2}{4}=1$(y−5)216−x24=1.
The graph of $\frac{x^2}{16}-\frac{y^2}{9}=1$x216−y29=1 is given below. Consider the translated hyperbola described by the equation $\frac{x^2}{16}-\frac{\left(y-2\right)^2}{9}=1$x216−(y−2)29=1.
What are the coordinates of the centre of the translated hyperbola?
What are the coordinates of the vertices of the translated hyperbola?
What are the coordinates of the foci of the translated hyperbola?
What are the equations of the asymptotes of the translated hyperbola?
Select the graph of the translated hyperbola, $\frac{x^2}{16}-\frac{\left(y-2\right)^2}{9}=1$x216−(y−2)29=1.
The graph of $\frac{x^2}{16}-\frac{y^2}{9}=1$x216−y29=1 is given below. Consider the translated hyperbola described by the equation $\frac{\left(x+3\right)^2}{16}-\frac{\left(y-7\right)^2}{9}=1$(x+3)216−(y−7)29=1.
What are the coordinates of the centre of the translated hyperbola?
What are the coordinates of the vertices of the translated hyperbola?
What are the coordinates of the translated hyperbola's focus?
What are the equations of the asymptotes of the translated hyperbola?
Select the graph of the translated hyperbola, $\frac{\left(x+3\right)^2}{16}-\frac{\left(y-7\right)^2}{9}=1$(x+3)216−(y−7)29=1.