Q97 / 160 · WBJEE 2012 · Mathematics

The transverse axis of a hyperbola is along the $x$ -axis and its length is $2 a$. The vertex of the hyperbola bisects the line segment joining the centre and the focus. The equation of the hyperbola is

  • A
    $6 x^{2}-y^{2}=3 a^{2}$
  • B
    $x^{2}-3 y^{2}=3 a^{2}$
  • C
    $x^{2}-6 y^{2}=3 a^{2}$
  • Correct answer
    D
    $3 x^{2}-y^{2}=3 a^{2}$

Solution

Let e be the eccentricity of hyperbola and length of conjugate axis be $2 h$. since, vertex $(a, 0)$ bisects the join of centre (0,0) and focus $(a e, 0)$ $$ \begin{array}{l} a=\frac{a e+0}{2} \Rightarrow e=2 \\ b^{2}=a^{2}\left(c^{2}-1\right)=3 a^{2} \end{array} $$ $\therefore$ Required hyperbola is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $\Rightarrow \quad \frac{x^{2}}{a^{2}}-\frac{y^{2}}{3 a^{2}}=1 \Rightarrow 3 x^{2}-y^{2}=3 a^{2}$

Hence, the correct answer is option: 'D'.

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