NIMCET 2017 — Mathematics PYQ

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Question

NIMCET 2017 — Mathematics PYQ

NIMCET | Mathematics | 2017

The equation of the hyperbola with center at the origin, length of the transverse axis is 6 and one focus at (0, 4) is?

Choose the correct answer:

Explanation

Concept:
The equation of the hyperbola is $\frac{y ^{2}}{b ^{2}} - \frac{x ^{2}}{a ^{2}} = 1$ with the foci $(0, \pm c)$

Length of the transverse axis = 2a

Calculations:
Since the coordinates of the one focus at $(0, 4) = (0, \pm c)$ , it is a case of vertical hyperbola
$\Rightarrow c = 2$
It is a case of vertical hyperbola

$\Rightarrow The equation of hyperbola is \frac{y ^{2}}{b ^{2}} - \frac{x ^{2}}{a ^{2}} = 1 ... (1)$
$\Rightarrow Length of the transverse axis = 6$
$\Rightarrow 2 a = 6$
$\Rightarrow a = 3$
$\Rightarrow c^{2} = a^{2} + b^{2}$
$\Rightarrow 2^{2} = 3^{2} + b^{2}$
$\Rightarrow b^{2} = 7$
$\Rightarrow Equation (1) becomes$
$\Rightarrow \frac{y ^{2}}{9} - \frac{x ^{2}}{7} = 1$
$\Rightarrow Hence, The equation of the hyperbola with center at the origin, length of the transverse axis is 6 and one focus at (0, 4) is \frac{y ^{2}}{9} - \frac{x ^{2}}{7} = 1$