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Rectangular Hyperbola - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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  • Rectangular Hyperbola is considered one of the most asked concept.

  • 36 Questions around this concept.

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The point of intersection of the curves whose parametric equations are \mathrm{x=t^{2}+1, y=2 t} and \mathrm{x}=2 \mathrm{~s}, \mathrm{y}=2 / \mathrm{s}, is given by

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\mathrm{e_{1}} and \mathrm{e_{2}} are the eccentricities of the hyperbolas \mathrm{x y=c^{2}} and \mathrm{x^{2}-y^{2}=c^{2}}, then \mathrm{e_{1}^{2}+e_{2}^{2}=}

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A circle cuts rectangular hyperbola xy = 1 in the points \mathrm{\left(x_r, y_r\right), r=1,2,3,4 \text {, }}  then 

 

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The foci of the ellipse \mathrm{\frac{x^2}{16}+\frac{y^2}{b^2}=1} and the hyperbola \mathrm{\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}} coincide, then the value of \mathrm{b^2} is

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The eccentricity of the hyperbola whose latus-rectum is 8 and conjugate axis is equal to half the distance between the foci, is 

 

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The equation of the hyperbola with vertices (3, 0) and (−3, 0) and semi latus rectum 4, is given by

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A tangent drawn to the hyperbola \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 } at \mathrm{ \quad P\left(\frac{\pi}{6}\right)} forms a triangle of area \mathrm{3 a^2 } sq. units with coordinate axes. The eccentricity of the hyperbola is equal to:

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If \mathrm{PQ} is a double ordinate of hyperbola \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1} such that \mathrm{CPQ} is an equilateral triangle, \mathrm{C} being the centre of the hyperbola.  Then the eccentricity e of the hyperbola satisfies

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The equation of the hyperbola whose foci are (6, 4) and (- 4, 4) and eccentricity 2 is given by

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If a circle cuts a rectangular hyperbola \mathrm{x y=c^2 \text { in } A, B, C, D} and the parameters of these four points be \mathrm{t_1, t_2, t_3 \text { and } t_4} respectively. Then

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Concepts Covered - 2

Rectangular Hyperbola

Rectangular Hyperbola

If the length of the transverse axis and the conjugate axis are equal (i.e. a = 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 \text { becomes } \mathrm{x}^{2}-\mathrm{y}^{2}=\mathrm{a}^{2}}

This is the general equation of a rectangular hyperbola.

 

Rectangular Hyperbola xy = c2

If we rotate the coordinate axes by 45o keeping the origin fixed, then the axes coincide with lines y = x and y = -x

\\\text{Using rotation, the equation } x^{2}-y^{2}=a^{2} \text { reduces to } \\ x y=\frac{a^{2}}{2}\\\\\Rightarrow xy = c^2

Properties of rectangular Hyperbola

For rectangular hyperbola, xy = c2

  1. Vertices: A(c, c) and A’(-c, -c) 

  2. Transverse axis: x = y

  3. Conjugate axis: x = -y

  4. Foci: S (c\sqrt{2},c\sqrt{2}) and  S' (-c\sqrt{2},-c\sqrt{2}) 

  5. Directrices: x + y = √2, x + y = - √2

  6. Length of latusrectum = AA’ = 2\sqrt{2}c

 

Properties of rectangular Hyperbola:

\\\mathrm{(i)\;\;\;\text{The parametric equation of the rectangular hyperbola}\;xy=c^2}\\\mathrm{\;\;\;\;\;\;\;\;\;are\;\;x=ct\;\;and\;\;y=\frac{c}{t}.}\\\\\mathrm{(ii)\;\;\;\text{The equation of the tangent to the rectangular hyperbola}\;xy=c^2}\\\mathrm{\;\;\;\;\;\;\;\;\;at\;(x_1,y_1)\;\;is\;\;x y_{1}+x_{1} y=2c^{2}.}\\\\\mathrm{(iii)\;\;\;\text { The equation of the tangent at }\left(c t, \frac{c}{t}\right) \text { to the hyperbola }\;xy=c^2}\\\mathrm{\;\;\;\;\;\;\;\;\;\text { is } \frac{x}{t}+y t=2 c.}\\\\\mathrm{(iv)\;\;\;\text { The equation of the normal at }\left(x_{1}, y_{1}\right) \text { to the hyperbola }\;xy=c^2}\\\mathrm{\;\;\;\;\;\;\;\;\;\text { is } x x_{1}-y y_{1}=x_{1}^{2}-y_{1}^{2}.}\\\\\mathrm{(v)\;\;\;\text { The equation of the normal at }t \text { to the hyperbola }\;xy=c^2}\\\mathrm{\;\;\;\;\;\;\;\;\;\text { is } x t^{3}-y t-c t^{4}+c=0.}

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Rectangular Hyperbola
Properties of rectangular Hyperbola

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