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Question

Find the equation of the hyperbola satisfying the give conditions: Vertices $$\displaystyle \left ( \pm  2,0 \right )$$ foci $$\displaystyle \left ( \pm  3,0 \right )$$


Solution

 Vertices $$\displaystyle \left ( \pm  2,0 \right )$$ foci $$\displaystyle \left ( \pm  3,0 \right )$$
Clearly the vertices are on the $$x$$-axis.
Therefore, the equation of the hyperbola is of the form  $$\displaystyle \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1$$
Since the vertices are $$\displaystyle \left ( \pm 2 , 0 \right )$$
$$\Rightarrow a = 2$$
and  foci are $$\displaystyle \left ( \pm 3,  o \right )\Rightarrow  ae = 3$$
We know that $$\displaystyle b^{2}= a^{2}(e^2-1)=a^2e^2-a^2=9-4=5$$
Thus the equation of the hyperbola is $$\displaystyle \frac{x^{2}}{4}-\frac{y^{2}}{5}= 1$$

Mathematics
NCERT
Standard XI

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