Question

If $(a^2,a-2)$ be a point interior to the region of the parabola $y^2=2x$ bounded by the chord joining the points $(2,2)$ and $(8,-4)$, then find the set of all possible real values of $a.$

Answer 1

Parabola: $y^2=2x............\left(1\right)$ $(a=\frac{1}{2})$

Equation of chord joining $(2,2)$ and $(8,-4)$, $(y-2)=\frac{-6}{6}(x+2)$

$=>y=-x+4.........\left(2\right)$

If point $P(a^2,a-2)$ is in the parabola then,

$S_1<0$

$=>(a-2{)}^2-2a^2<0$

$=>-a^2-4a+4<0$

$=>a^2+4a-4>0$

$=>(a+2+2\sqrt{2})(a+2-2\sqrt{2})>0$

$=>aϵ(-\infty ,-2-2\sqrt{2})\cup (-2+2\sqrt{2},\infty )....................\left(3\right)$

Also it should be in the region bounded by $\left(1\right)$ and $\left(2\right)$, Point $P(a^2,a-2)$ must be on the side of origin so, $\frac{{ax}_1+{by}_1+c}{{ax}_2+{by}_2+c}>0$ where

$(x_1,y_1):(a^2,a-2)$ and $(x_2,y_2):(0,0)=>\frac{a^2+a-2-4}{0+0-4}>0$

$=>a^2+a-6>0$

$=>(a+3)(a-2)>0$

$=>aϵ(-\infty ,-3)\cup (2,\infty )...............\left(2\right)$

Intersection of $\left(3\right)$ and $\left(4\right)$,

$=>aϵ(-\infty ,-2-2\sqrt{2})\cup (-2+2\sqrt{2},\infty ).$