Login/Sign Up
Home » Find the equations of the hyperbola satisfying the given conditions. Vertices (±7, 0), e = 4/3
Find the equations of the hyperbola satisfying the given conditions. Vertices (±7, 0), e = 4/3
CBSE | Class 11 | Conic Sections | Exercise 11.4 | Maths | NCERT
Find the equations of the hyperbola satisfying the given conditions. Vertices (±7, 0), e = 4/3

Given:

    \[Vertices\text{ }\left( \pm 7,\text{ }0 \right)\text{ }and\text{ }e\text{ }=\text{ }4/3\]

Here, the foci are on x-axis.

The condition of the hyperbola is of the structure

    \[{{x}^{2}}/{{a}^{2}}~\text{ }{{y}^{2}}/{{b}^{2}}~=\text{ }1\]

Since, the foci are

    \[~\left( \pm ~7,\text{ }0 \right),~\]

along these lines,

    \[a\text{ }=\text{ }7\]

    \[\begin{array}{*{35}{l}} It\text{ }is\text{ }given\text{ }that\text{ }e\text{ }=~4/3 \\ c/a\text{ }=\text{ }4/3 \\ 3c\text{ }=\text{ }4a \\ \end{array}\]

    \[\begin{array}{*{35}{l}} Substitute\text{ }the\text{ }value\text{ }of\text{ }a,\text{ }we\text{ }get \\ 3c\text{ }=\text{ }4\left( 7 \right) \\ c\text{ }=\text{ }28/3 \\ \end{array}\]

It is realized that,

    \[{{a}^{2}}~+\text{ }{{b}^{2}}~=\text{ }{{c}^{2}}\]

    \[<span class="ql-right-eqno"> (1) </span><span class="ql-left-eqno"> </span><img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-95d3854666f9cab18d2437a680a22f1a_l3.png" height="168" width="279" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*} & \begin{array}{*{35}{l}} {{7}^{2}}~+\text{ }{{b}^{2}}~=\text{ }{{\left( 28/3 \right)}^{2}} \\ {{b}^{2}}~=\text{ }784/9\text{ }\text{ }49 \\ \end{array} \\ & \begin{array}{*{35}{l}} =\text{ }\left( 784\text{ }\text{ }441 \right)/9 \\ =\text{ }343/9 \\ \end{array} \\ \end{align*}" title="Rendered by QuickLaTeX.com"/>\]

∴ The condition of the hyperbola is

    \[{{x}^{2}}/49\text{ }\text{ }9{{y}^{2}}/343\text{ }=\text{ }1\]

i

More Material