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Home » Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the following cases: (i) the distance between the foci = 16 and eccentricity = √2 (ii) conjugate axis is 5 and the distance between foci = 13
Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the following cases: (i) the distance between the foci = 16 and eccentricity = √2 (ii) conjugate axis is 5 and the distance between foci = 13
CBSE | Class 11 | Exercise 27.1 | Hyperbola | Maths | RD Sharma
Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the following cases: (i) the distance between the foci = 16 and eccentricity = √2 (ii) conjugate axis is 5 and the distance between foci = 13

(i) the distance between the

    \[foci\text{ }=\text{ }16\text{ }and\text{ }eccentricity\text{ }=\text{ }\surd 2\]

Given:

Distance between the foci

    \[=\text{ }16\]

Eccentricity

    \[=\text{ }\surd 2\]

Let us compare with the equation of the form

RD Sharma Solutions for Class 11 Maths Chapter 27 – Hyperbola - image 51

Distance between the foci is

    \[2ae\text{ }and\text{ }{{b}^{2}}~=\text{ }{{a}^{2}}({{e}^{2}}-\text{ }1)\]

So,

    \[2ae\text{ }=\text{ }16\]

    \[ae\text{ }=\text{ }16/2\]

Or,

    \[a\surd 2\text{ }=\text{ }8\]

    \[a\text{ }=\text{ }8/\surd 2\]

Or,

    \[{{a}^{2}}~=\text{ }64/2\]

    \[=\text{ }32\]

We know that,

    \[{{b}^{2}}~=\text{ }{{a}^{2}}({{e}^{2}}-\text{ }1)\]

So,

    \[{{b}^{2}}~=\text{ }32\text{ }[{{\left( \surd 2 \right)}^{2}}-\text{ }1]\]

    \[=\text{ }32\text{ }\left( 2\text{ }-\text{ }1 \right)\]

Hence,

    \[=\text{ }32\]

The Equation of hyperbola is given as

RD Sharma Solutions for Class 11 Maths Chapter 27 – Hyperbola - image 52

    \[{{x}^{2}}-\text{ }{{y}^{2}}~=\text{ }32\]

∴ The Equation of hyperbola is

    \[{{x}^{2}}-\text{ }{{y}^{2}}~=\text{ }32\]

(ii) conjugate axis is

    \[5\]

and the distance between foci

    \[=\text{ }13\]

Given:

Conjugate axis

    \[=\text{ }5\]

Distance between foci

    \[=\text{ }13\]

Let us compare with the equation of the form

RD Sharma Solutions for Class 11 Maths Chapter 27 – Hyperbola - image 53

Distance between the foci is

    \[2ae\text{ }and\text{ }{{b}^{2}}~=\text{ }{{a}^{2}}({{e}^{2}}-\text{ }1)\]

Length of conjugate axis is

    \[2b\]

So,

    \[2b\text{ }=\text{ }5\]

    \[b\text{ }=\text{ }5/2\]

Or,

    \[{{b}^{2}}~=\text{ }25/4\]

We know that,

    \[2ae\text{ }=\text{ }13\]

    \[ae\text{ }=\text{ }13/2\]

Or,

    \[{{a}^{2}}{{e}^{2}}~=\text{ }169/4\]

    \[{{b}^{2}}~=\text{ }{{a}^{2}}({{e}^{2}}-\text{ }1)\]

Or,

    \[{{b}^{2}}~=\text{ }{{a}^{2}}{{e}^{2}}-\text{ }{{a}^{2}}\]

    \[25/4\text{ }=\text{ }169/4\text{ }-\text{ }{{a}^{2}}\]

Or,

    \[{{a}^{2}}~=\text{ }169/4\text{ }-\text{ }25/4\]

    \[=\text{ }144/4\]

So,

    \[=\text{ }36\]

The Equation of hyperbola is given as

RD Sharma Solutions for Class 11 Maths Chapter 27 – Hyperbola - image 54

RD Sharma Solutions for Class 11 Maths Chapter 27 – Hyperbola - image 55

∴ The Equation of hyperbola is

    \[25{{x}^{2}}~\text{ }144{{y}^{2}}~=\text{ }900\]

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