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Rectangular Hyperbola

As we know that an ellipse is a special type of circle; likewise rectangular hyperbola is a special type of hyperbola.

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Posted by Kunduz Tutor, 18/11/2021
Rectangular Hyperbola

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As we know that an ellipse is a special type of circle; likewise rectangular hyperbola is a special type of hyperbola. There are a few special types of hyperbola(s), out of which one is the rectangular hyperbola. To define a rect. hyperbola, the number of equivalent ways is more than one.

Author – Ojasvi Chaplot


Rectangular Hyperbola Definition

A rectangular hyperbola can be described as the hyperbola whose asymptotes are at right angles to each other. 

Rectangular Hyperbola Equation

The angle between the asymptotes of the hyperbola: Rectangular hyperbola,  is given as:

So, this is a right-angle,

So, both the axes, the transverse axis and the conjugate axis are equal. Therefore, the equation of rect. hyperbola is written as: Equation of rectangular hyperbola

Hence, the equation of rectangular hyperbola is:

Equation of rectangular hyperbola 1


Eccentricity of Rectangular Hyperbola

Since, the eccentricity of the hyperbola is obtained by:

Then, the eccentricity of rectangular hyperbola is:

Eccentricity of rectangular hyperbola

Rect. Hyperbola is a special type of hyperbola in the conic section, the eccentricity of which is equal to √2. Rect. hyperbola is also referred to as equilateral hyperbola.

Note – It can also be called as square hyperbola. Let us explain why we can call it square. When the asymptotes of the hyperbola are drawn and then a rectangle is drawn with the help of those asymptotes, so we will observe that the length is equal to the breadth of that rectangle in a few cases. 


Rectangular Hyperbola Graph

Rectangular hyperbola graph
A Rectangular Hyperbola

Important Points of Rectangular Hyperbola

  • In a hyperbola, since b2 = a2 (e2 – 1). In the case of rect. hyperbola (i.e., when b = a), it becomes
    • a2 = a2(e2 – 1)
    • e2 = 2
    • e = √2
    • i.e. the eccentricity of a rect. hyperbola is equal to √2.
  • In case of the rect. hyperbola, since a = b, therefore the length of the transverse axis is equal to the length of the conjugate axis.
  • A rect. hyperbola is also referred as an equilateral hyperbola.
  • The asymptotes of the rect. hyperbola are y = ± x.

Special Case

If the axes of the hyperbola are rotated by an angle of (-π/4) about the same origin, then the equation of the rect. hyperbola x2 – y2 = ais reduced to xy = c2; where c2 = a2/2, which can be illustrated as shown:

So, for this we have to write [(x/√2 + y/√2) for x] and [(–x/√2 + y/√2) for y].

Then, the equation becomes

  • When the equation of the rect. hyperbola is xy = c2, then the asymptotes of it are the coordinate axes.
  • The length of the latus rectum of rectangular hyperbola is the same as that of the length of transverse axis or conjugate axis.

The rect. hyperbola with its asymptotes as the coordinate axes states the following properties and characteristics: 

  • The equation of the asymptotes of the rectangular hyperbola are: x = 0 and y = 0.
  • The equation of the rect. hyperbola is xy = c2 and with the parametric representation as: x = ct and y = c/t; where, t ∈ R – {0}. So, the parametric point is (ct, c/t).
  • The equation of the transverse axis of the rectangular hyperbola is: y = x.
  • The equation of the conjugate axis of the rectangular hyperbola is: y = -x.
  • The center of the rectangular hyperbola is at (0, 0).
  • The vertices of the rectangular hyperbola are (c, c) and (-c, -c).
  • The foci of the rectangular hyperbola are (√2c, √2c) and (-√2c, -√2c).
  • The equations of the directrices of the rect. hyperbola are: x + y = ± √2c.
  • Since, the transverse axis and the conjugate axis are same.Therefore, the length of latus rectum = 2√2c = length of T.A. = length of C.A
  • The equation of the chord to the rect. hyperbola whose mid-point is given as (p, q) is: qx + py = 2pq.
  • The equation of the tangent to the rect. hyperbola at a point P(x1, y1) is: x/x1 + y/y1 = 2.
  • And, at parametric point P(t) i.e. P(ct, c/t), the equation of the tangent is: x/t + ty = 2c.
  • The equation of the normal to the rect. hyperbola is: y – c/t = t2(x – ct).
  • The equation of the chord to the rect. hyperbola, joining the points P(t1) and Q(t2) is written as: x + (t1t2)y = c(t1 + t2) and its slope, m = -1/t1t2.  
  • A rect. hyperbola which circumscribes a triangle, passes through the orthocentre of the triangle.
  • If a circle intersects a rect. hyperbola at four points, then the mean value of the points of intersection is the mid-point of the line joining the centres of both, i.e. the circle and the hyperbola.

Solved Example on Rectangular Hyperbola

Example-1: Find out the vertices and the foci of a rectangular hyperbola whose equaton is given as xy = 1.

Solution: 

Given that, xy = 1.

Clearly, the given equation of the rect. hyperbola is of the form xy = c.

So, the vertices of the rect. hyperbola [xy = c] are (c, c) and (-c, -c).

  • (1, 1) and (-1, -1)

 And, the foci of the rect. hyperbola [xy = c] are (√2c, √2c) and (-√2c, -√2c).

  • (√2, √2) and (-√2, -√2)

Answer: The vertices of the rect. hyperbola are (1, 1) and (-1, -1) and the foci of the rect. hyperbola are (√2, √2) and (-√2, -√2).

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