While studying hyperbola, I came across a question:
Let $y=mx+c$ is a tangent to a hyperbola $$\cfrac{x^2}{ \lambda^2} -\cfrac{y^2}{( \lambda^3+ \lambda^2+\lambda)^2} = 1$$Find least value of $16m^2$.
My attempt:
As $y =mx +c$ is tangent so $c^2=a^2m^2-b^2$ then I put value of $a$ and $b$ and I take derivative of it but there is no information about $\lambda$. How should I proceed?
As you have said $c^2 = a^2 m^2 - b^2$ ,On placing values$$ c = \pm \sqrt{\lambda^2 m^2 - (\lambda + \lambda^2 + \lambda^3)^2}$$ $$ \implies {\lambda^2 m^2 - (\lambda + \lambda^2 + \lambda^3)^2} \ge 0 $$ $$ \implies \lambda^2( m^2 - (1 + \lambda + \lambda^2)^2 )\ge 0 $$ Assuming $ \lambda \ne 0$, $$ m^2 \ge ( 1 + \lambda + \lambda^2)^2 $$ As nothing is specified about the nature of values of $ \lambda$, it can be assumed that $\lambda \in \mathbb R - \{0\}$. Least value of the polynomial $ 1 + \lambda + \lambda^2$ is $\dfrac{3}{4}$ $$ \implies m^2 \ge \dfrac{9}{16} \implies 16 m^2 \ge 9 $$