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Geometry Level 5

If two tangents can be drawn to the different branches of hyperbola $x^{2}-\dfrac{y^{2}}{4}=1$ from the point $(\omega,\omega^{2})$ then $\omega$ belongs to $\text{\_\_\_\_\_\_\_\_\_\_\_\_\_}$ .

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(0,2)

(-2,0)

(-\infty,-1)

(2,\infty)

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Archit Tripathi
Oct 26, 2016

$(t,t^{2})$ lies on the parabola $y = x^{2}$ .

$(t,t^{2})$ must lie between the asymptotes of hyperbola in first and second quadrant.

Since asymptotes are $y = 2x , y = -2x$

$\Rightarrow$ $2t < t^{2}$

$\Rightarrow$ $t < 0$ or $t > 2$

and $-2t < t^{2}$

$\Rightarrow$ $t < -2$ or $t > 0$

On combining these results we have,

$t < -2$ or $t > 2$

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