Question

If two tangents can be drawn to the different branches of hyperbola
$\frac{x^2}{1}-\frac{y^2}{4}=1$ from the paint $(\alpha ,{\alpha }^2)$, then

• A. $\alpha ϵ(-2,0)$
  
• B. $\alpha ϵ(-3,0)$
• C. $\alpha ϵ(-\infty ,-2)$
  
• D. $\alpha ϵ(-\infty ,-3)$

Answer 1

The correct option is C $\alpha ϵ(-\infty ,-2)$


Given that,
$\frac{x^2}{1}-\frac{y^2}{4}=1$




Since, $(\alpha ,{\alpha }^2)$ lie on the parabola $y=x^2$, then $(\alpha ,{\alpha }^2)$ must lie between the asymptotes of hyperbola $\frac{x^2}{1}-\frac{y^2}{4}=1$ in 1st and 2nd quadrants.
So, the asymptotes are $y=\pm 2x$
$\therefore$ $2\alpha <{\alpha }^2$
$\Rightarrow$ $\alpha <0or\alpha >2$ and $-2\alpha <{\alpha }^2$
$\alpha <-2or-2\alpha <0$
$\therefore$ $\alpha ϵ(-\infty ,-2)or(2,\infty )$