Question

Prove that $xy={{c}^{2}}$ is another form of rectangular hyperbola.

Answer 1

Using the definition of rectangular hyperbola, we can derive an equation for rectangular hyperbola. Then, we will need to rotate the coordinate axes by $-{{45}^{\circ }}$, so that we can change the derived equation into the required form.

Complete step by step answer:
We know that if the asymptotes of a hyperbola are perpendicular to each other, then the hyperbola is called rectangular hyperbola. Also, the lengths of semi major axis and semiminor axes are equal to each other.
We know that the general equation of a hyperbola is $\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1$.
Now, since in the rectangular hyperbola, the length of the semimajor and semiminor axes are the same, we can write a = b.
Thus, the equation for a rectangular hyperbola becomes,
${{x}^{2}}-{{y}^{2}}={{a}^{2}}...\left( i \right)$.
We can also write the above equation as
$\left( x+y \right)\left( x-y \right)={{a}^{2}}$
The two asymptotes for this parabola are $x+y=0$ and $x-y=0$. These two asymptotes are at an angle of ${{135}^{\circ }}\text{ and 4}{{\text{5}}^{\circ }}$ with the positive X-axis.
Let us now rotate the whole system by an angle of $-{{45}^{\circ }}$.
We know that for a rotation of angle $\theta$, $x$ is replaced by $\left( x\cos \theta -y\sin \theta \right)$ and $y$ is replaced by $\left( x\sin \theta +y\cos \theta \right)$. So, for a rotation by an angle of $-{{45}^{\circ }}$, $x$ will be replaced by $\left( \dfrac{x+y}{\sqrt{2}} \right)$ and $y$ will be replaced by $\left( \dfrac{-x+y}{\sqrt{2}} \right)$.
Thus, with the help of equation (i), we can say that the equation for rectangular parabola can also be written as,
${{\left( \dfrac{x+y}{\sqrt{2}} \right)}^{2}}-{{\left( \dfrac{-x+y}{\sqrt{2}} \right)}^{2}}={{a}^{2}}$
We can simplify the above equation as
$\dfrac{1}{2}\left[ \left( {{x}^{2}}+{{y}^{2}}+2xy \right)-\left( {{x}^{2}}+{{y}^{2}}-2xy \right) \right]={{a}^{2}}$
On further simplification, we get
$2xy={{a}^{2}}$
We can also write the above equation as
$xy={{c}^{2}}$, where ${{c}^{2}}=\dfrac{{{a}^{2}}}{2}$.
Thus, we have proved that $xy={{c}^{2}}$ is another form of rectangular hyperbola.

Note: We must remember that a rectangular hyperbola is also known as equilateral hyperbola or right hyperbola. Also, it is useful to remember that the eccentricity of rectangular hyperbola is $\sqrt{2}$. We also know that the asymptotes of $xy={{c}^{2}}$ will be the coordinate axes.