Question

If ax+by+c=0 and lx+my+n=0 are asymptotes of a hyperbola, then

• A. am≠bl
• B. $\frac{am+bl}{al+bm}>4$
• C. al≠bm
• D. none of these

Answer 1

The correct option is A am≠bl

Equation of a hyperbola whose asymptotes are $ax+by+c=0$ and $b+my+n=0$ can be written as $(ax+by+c)(1x+my+n)=\lambda$

i.e. $al{x}^2+(am+bl)xy+bm{y}^2+(an+cl)x+(bn+cm)y+cn-\lambda =0$

which will represent a hyperbola if ${\left(\frac{am+bl}{2}\right)}^2-albm>0$

i.e. $(am+bl{)}^2-4(am\left)\right(bl)>0$
i.e. $(am-bl{)}^2>0$
i.e. $am\ne bl$