Rectangular Hyperbola: Equation, Graph, Questions, Examples

Rectangular Hyperbola is the Hyperbola in which the length of the transverse axis and the conjugate axis are equal (i.e. a = b). The asymptote of the rectangular hyperbola is y = ±x. Also, the asymptotes of a rectangular hyperbola are perpendicular. In real life, we use Rectangular Hyperbolas, used for predicting the path of the satellite.

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This Story also Contains

1. What is Rectangular Hyperbola?
2. Rectangular Hyperbola Equation
3. Properties of rectangular Hyperbola
4. Solved Examples Based on Rectangular Hyperbola
5. Summary

In this article, we will cover the concept of the Rectangular Hyperbola. This category falls under the broader category of Coordinate Geometry, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. A total of thirty questions have been asked on JEE MAINS( 2013 to 2023) from this topic.

What is Rectangular Hyperbola?

Hyperbola in which the length of the transverse axis and the conjugate axis are equal (i.e. $\mathrm{a}=\mathrm{b}$ ) then the hyperbola is known as a rectangular hyperbola or equilateral hyperbola. The eccentricity of the rectangular hyperbola is $\sqrt{2 }$. The length of the transverse axis $2 \mathrm a$ and the length of the conjugate axis $2 \mathrm b$ are equal.

Rectangular Hyperbola Shape

The shape of a rectangular hyperbola is defined by two distinct branches that extend indefinitely, bending away from each other in opposing quadrants. Each branch mirrors the other across the origin when the hyperbola is centred at the origin.

Rectangular Hyperbola Equation

The general equation of the Rectangular Hyperbola centered at the origin $(0,0)$ is, $\mathrm{a}=\mathrm{b}$

So, $\frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$ becomes $\mathrm{x}^2-\mathrm{y}^2=\mathrm{a}^2$
If the centre shifts to $\left(x_0, y_0\right)$, then the equation of rectangular hyperbola becomes
$
\left(x-x_0\right)^2-\left(y-y_0\right)^2=a^2
$

Parametric Equation of Rectangular Hyperbola

The parametric equation of the rectangular hyperbola is,
$
\begin{aligned}
& x=a \sec \theta \\
& y=a \tan \theta
\end{aligned}
$

Rectangular Hyperbola Graph

A rectangular hyperbola is a type of hyperbola that is specifically defined as having the property that the asymptotes are perpendicular to each other, forming a right angle. Graph of a Rectangular Hyperbola with equation $x y=c^2$ where $c$ is a constant that determines the scale of the hyperbola.

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
$
\begin{aligned}
& x y=\frac{a^2}{2} \\
& \Rightarrow x y=c^2
\end{aligned}
$

For rectangular hyperbola, $x y=c^2$

1. Vertices: $\mathrm A(c, c)$ and $\mathrm A^{\prime}(-c,-c)$
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 latus rectum $=\mathrm{AA}^{\prime}=2 \sqrt{2} c$

Eccentricity of Rectangular Hyperbola

The equation of the rectangular hyperbola is $x^2-y^2=a^2$. Now we know that the eccentricity of the hyperbola is,
$
e=\sqrt{1+\frac{b^2}{a^2}}
$

In the case of rectangular hyperbola
$
\begin{aligned}
& a=b \Rightarrow b^2=a^2 \\
\Rightarrow &\space a^2\left(e^2-1\right)=a^2 \\
\Rightarrow & \space e=\sqrt{2}
\end{aligned}
$

Properties of rectangular Hyperbola

(i) The parametric equation of the rectangular hyperbola $\mathrm{xy}=\mathrm{c}^2$ are $x=$ ct and $y=\frac{c}{t}$.
(ii) The equation of the tangent to the rectangular hyperbola $x y=c^2$ 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 $x y=c^2$ is $\mathrm{xt}^3-\mathrm{yt}-\mathrm{ct}{ }^4+\mathrm{c}=0$.

Asymptotes of a Rectangular Hyperbola

Asymptotes are the lines that connect the curve at infinity. In the case of rectangular hyperbola, the equation of asymptote is,
$
\begin{aligned}
& y= \pm x \\
& x^2-y^2=0
\end{aligned}
$

Asymptotes of a Rectangular Hyperbola are Perpendicular.

Recommended Video Based on Rectangular Hyperbola

Solved Examples Based on Rectangular Hyperbola

Example 1: If the equation $4 \mathrm{x}^2+\mathrm{ky}^2=18$ represents a rectangular hyperbola, then k is equal to
1) 4
2) -4
3) 3
4) None of these

Solution:
Clearly for $4 \mathrm{x}^2+\mathrm{ky}^2=18$ to represent a rectangular hyperbola $\mathrm{k}=4$
Hence, the answer is the option 1.

Example 2: At the point of intersection of the rectangular hyperbola $\mathrm{xy}=\mathrm{c}^2$ and the parabola $\mathrm{y}^2=4 \mathrm{ax}$. The tangents to the rectangular hyperbola and the parabola make an angle $\theta$ and $\phi$ respectively with the axis of X, then

1) $\theta=\tan ^{-1}(-2 \tan \phi)$
2) $\phi=\tan ^{-1}(-2 \tan \theta)$
3) $\theta=\frac{1}{2} \tan ^{-1}(-\tan \phi)$
4) $\phi=\frac{1}{2} \tan ^{-1}(-\tan \theta)$

Solution:

Let $\left(x_1, y_1\right)$ be the point of intersection $\Rightarrow y_1^2=4 a x_1$ and $x_1 y_1=c^2$

For Parabola we have,
$
\begin{aligned}
& y^2=4 a x \\
& \therefore \frac{d y}{d x}=\frac{2 a}{y} \\
& \frac{d y}{d x}{ }_{\left(x_1, y_1\right)}=\tan \phi=\frac{2 a}{y_1}
\end{aligned}
$

For rectangular hyperbola we have,
$
\begin{aligned}
& x y=c^2 \\
& \frac{d y}{d x}=-\frac{y}{x} \\
& \frac{d y}{d x}\left(x_1, y_1\right)=\tan \phi=\frac{y_1}{x_1} \\
& \therefore \frac{\tan \theta}{\tan \phi}=\frac{-y_1 / x_1}{2 a / y_1}=\frac{-y_1^2}{2 a x_1}=-\frac{4 a_1}{2 a_1}=-2 \\
& \Rightarrow \theta=\tan ^{-1}(-2 \tan \phi)
\end{aligned}
$

Hence, the answer is option 1.

Example 3: Find the foci of the rectangular hyperbola whose equation is $x^2-y^2=16$.
Solution:
Equation of Rectangular Hyperbola is, $x^2-y^2=a^2 \ldots$ (i)
Given Equation,
$x^2-y^2=16$.
$
x^2-y^2=4^2
$

Comparing Equation (i) and (ii)
$
a=4
$

Foci of Rectangular Hyperbola is $( \pm a \sqrt{ 2},0$ )
So, Foci of Given Rectangular Hyperbola is $( \pm 4 \sqrt{2},0)$

Example 4: If tangents $OQ$ and $OR$ from $O$ are drawn to a variable circle having radius r and the centre lying on the rectangular hyperbola $x y=1$, then the locus of circumcentre of $\triangle O Q R$ is equal to... ( $O$ is the origin)

Solution:

Let $S\left(t, \frac{1}{t}\right)$ be any point on the given rectangular hyperbola $x y=1$.

A circle is drawn with a centre at $S$ and radius $r$. From origin $O$ tangents $O Q$ and $O R$ are drawn to the above circle. $O Q S R$ is a cyclic quadrilateral.
Hence, points $O, Q, S$ and $R$ are concyclic.

The Circumcircle of $\triangle O Q R$ also passes through $S$ and $O S$ is the diameter.
Therefore, the circumcentre of $\triangle O Q R$ is the mid-point of $O S$. If $(x, y)$ is the circumcentre of $\triangle O Q R$, then

$x=\frac{0+t}{2}, y=\frac{0+\frac{1}{t}}{2}$

$\therefore x y=\frac{1}{4}$

So, the required locus is $x y=\frac{1}{4}$.
Hence, the answer is $x y=\frac{1}{4}$

Example 5: Consider the set of hyperbolas $x y=k, x \in R$. Let $e_1$ be the eccentricity when $k=4$ and $e_2$ be the eccentricity when $k=9$, then $e_1-e_2$ is equal to:
Solution:
We know that the eccentricity of $x y=k$ for all $k \in R$ is $\sqrt{2}$.
$\therefore \quad e_1=\sqrt{2}$ and also $e_2=\sqrt{2}$
Hence,
$
e_1-e_2=0
$

Hence, the required answer is 0.

Summary

A rectangular hyperbola is a special type of hyperbola defined by its unique symmetry and perpendicular asymptotes. Its unique equation and asymptotic behaviour make it a fundamental concept in mathematics, applied sciences, and optics. Knowledge of Hyperbola helps us to solve complex problems.