Rectangular Hyperbola - Practice Questions & MCQ

Rectangular Hyperbola

If the length of the transverse axis and the conjugate axis are equal (i.e. $\mathrm{a}=\mathrm{b}$ ) then the hyperbola is known as rectangular hyperbola or equilateral hyperbola.

$
\mathrm{a}=\mathrm{b}
$

So, $\frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$ becomes $\mathrm{x}^2-\mathrm{y}^2=\mathrm{a}^2$
This is the general equation of a rectangular hyperbola.

Rectangular Hyperbola $x y=c^2$
If we rotate the coordinate axes by $45^{\circ}$ keeping the origin fixed, then the axes coincide with lines $y=x$ and $y=-x$
Using rotation, the equation $x^2-y^2=a^2$ reduces to
$x y=\frac{a^2}{2}$
$\Rightarrow x y=c^2$

For rectangular hyperbola, xy = c²

1. 1. Vertices: $A(c, c)$ and $A^{\prime}(-c,-c)$
   2. Transverse axis: $x=y$
   3. Conjugate axis: $x=-y$
   4. Foci: $\mathrm{S}(c \sqrt{2}, c \sqrt{2})$ and $\mathrm{S}^{\prime}(-c \sqrt{2},-c \sqrt{2})$
   5. Directrices: $x+y=\sqrt{2}, x+y=-\sqrt{2}$
   6. Length of latusrectum $=A A^{\prime}=2 \sqrt{2} c$

Properties of rectangular Hyperbola:
(i) The parametric equation of the rectangular hyperbola $x y=c^2$ are $\mathrm{x}=\mathrm{ct}$ and $\mathrm{y}=\frac{\mathrm{c}}{\mathrm{t}}$.
(ii) The equation of the tangent to the rectangular hyperbola $\mathrm{xy}=\mathrm{c}^2$ at $\left(\mathrm{x}_1, \mathrm{y}_1\right)$ is $\mathrm{xy}_1+\mathrm{x}_1 \mathrm{y}=2 \mathrm{c}^2$.
(iii) The equation of the tangent at $\left(\mathrm{ct}, \frac{\mathrm{c}}{\mathrm{t}}\right)$ to the hyperbola $\mathrm{xy}=\mathrm{c}^2$ is $\frac{\mathrm{x}}{\mathrm{t}}+\mathrm{yt}=2 \mathrm{c}$.
(iv) The equation of the normal at $\left(\mathrm{x}_1, \mathrm{y}_1\right)$ to the hyperbola $\mathrm{xy}=\mathrm{c}^2$ is $\mathrm{xx}_1-\mathrm{yy}_1=\mathrm{x}_1^2-\mathrm{y}_1^2$.
(v) The equation of the normal at $t$ to the hyperbola $\mathrm{xy}=\mathrm{c}^2$ is $\mathrm{xt}^3-\mathrm{yt}-\mathrm{ct}^4+\mathrm{c}=0$.