Question

Determine all the values of $x$ for which the points $(\alpha ,{\alpha }^2)$ lies inside the $△$ formed by the lines
$x-3y+2=0$
$x-2y-3=0$
$x-5y+6=0$

Answer 1

It is given that point $P({\alpha }^2,\alpha )$ lies either inside or on the triangle

So, A and P must lie on same side of BC.

$\Rightarrow (9-9+2)({\alpha }^2-3\alpha +2)\ge 0\Rightarrow (\alpha -2)(\alpha -1)\ge 0$

$\Rightarrow \alpha \in (-\infty ,1]\cup [2,\infty )$ __(i)

if B and P lie on the same side of AC then

$(4-4-3)({\alpha }^2-2\alpha -3)\ge 0$

$\Rightarrow (\alpha -3)(\alpha +1)\ge 0$

$\Rightarrow \alpha \in [-1,3]$ __(ii)

if C and P lie on the same side of AB , then

$(13-25+6)({\alpha }^2-52+6)\ge 0$

$\Rightarrow (\alpha -3)(\alpha -2)\le 0$

$\Rightarrow \alpha \in [2,3]$ __(iii)

From (i), (ii), (iii)

$\alpha \in [2,3]$