Question
The equation of a parabola is $${y^2} = 4x.\,P\left( {1,\,3} \right)$$ and $$Q\left( {1,\,1} \right)$$ are two points in the $$x$$-$$y$$ plane. Then, for the parabola :
A.
$$P$$ and $$Q$$ are exterior points
B.
$$P$$ is an interior point while $$Q$$ is an exterior point
C.
$$P$$ and $$Q$$ are interior points
D.
$$P$$ is an exterior point while $$Q$$ is an interior point
Answer :
$$P$$ is an exterior point while $$Q$$ is an interior point
Solution :
$$\eqalign{
& {\text{Here, }}S \equiv {y^2} - 4x = 0 \cr
& S\left( {1,\,3} \right) = {3^2} - 4.1 > 0\,\,\,\,\,\, \Rightarrow P\left( {1,\,3} \right){\text{ is an exterior point}}{\text{.}} \cr
& S\left( {1,\,1} \right) = {1^2} - 4.1 < 0\,\,\,\,\,\, \Rightarrow Q\left( {1,\,1} \right){\text{ is an interior point}}{\text{.}} \cr} $$