Question

Consider the parabola $y=ax–b{x}^2$. If the least positive value of a for which there exist $\alpha ,\alpha ϵR–\left\{0\right\}$ such that both the point $(\alpha ,\beta )$ and $(\beta ,\alpha )$ lies on the given parabola is k then [k] is equal to ___

Answer 1

$(\alpha ,\beta )$ and $(\beta ,\alpha )$ lie on some line $y=–x+\lambda$

Solving line and parabola: $(–x+\lambda )=ax–b{x}^2$

$\Rightarrow b{x}^2–(1+a)x+\lambda =0\dots \left(1\right)$

Again, $y=a(\lambda –y)–b(\lambda –y{)}^2$

$\Rightarrow b{y}^2+y(1+a-2b\lambda )+(b{\lambda }^2–a\lambda )=0\dots \left(2\right)$

Both (1) and (2) has same roots which are $\alpha$ and $\beta$

Hence, $1+a-2b\lambda =–(1+a)\Rightarrow \lambda =\frac{(1+a)}{b}$

Now, for $\alpha ,\beta$ to exist discriminant of (1) > 0

$\Rightarrow (1+a{)}^2–4b\lambda >0$

$\Rightarrow (a–3)(a+1)>0$

$\Rightarrow a\in (–\infty ,–1)\cup (3,\infty )$