The equation to the hyperbola of given transverse axis whose vertex bisects the distance between the centre and focus, is given by

3 x^{2} - y^{2} = 3 a^{2}

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x^{2} - 3 y^{2} = a^{2}

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x^{2} - y^{2} = 3 a^{2}

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None of these

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Solution

The correct option is A $3 x^{2} - y^{2} = 3 a^{2}$
Let the equation of hyperbola be $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ..... $(i)$
It is given that the vertex bisects the distance between centre and focus of the hyperbola
$∴ a = a e - a ⟹ 2 a = a e ⟹ e = 2$
Length of transverse axis is $2 a$
Now, $b^{2} = a^{2} (e^{2} - 1) = a^{2} (4 - 1) = 3 a^{2}$
Equation $(i)$ becomes
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{3 a^{2}} = 1$
$⟹ 3 x^{2} - y^{2} = 3 a^{2}$

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Q. If the tangent at the point

(a sec ϕ, b tan ϕ)

to the hyperbola

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

meets the transverse axis at

T

, then the distance of

T

from the focus of the hyperbola is :

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