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 the set of all possible real values of a is

Options

A. `(-2, sqrt2)`
B. `(-3, 2)`
C. `(-2, 2sqrt2)`
D. `(-2, -2+sqrt2)`

Answer

D

Explanation

If `(a^(2), a-2)` lies inside the parabola, then `(a-2)^(2)-2xxa^(2)lt0`
`rArr" "a^(2)-4+4-2a^(2)lt0`
`rArr" "-a^(2)-4a+4lt0`
`rArr" "a^(2)+4a-4gt0`
`rArr" "(a+3)^(2)-(2sqrt2)^(2)gt0`
`rArr" "a+2lt-2sqrt2or, a+2gt2sqrt2`
`rArr" "alt-2-2sqrt2or, agt2sqrt2-2" ...(i)"`

Since point `P(a^(2), a-2)` and the origin O (0, 0) are on the same side of chord joining (2, 2) and (8, -4).
`:." "(0+0-4)(a^(2)+a-2-4)gt0`
`rArr" "a^(2)+a-6lt0rArr(a+3)(a-2)lt0rArr-3ltalt2" ....(ii)"`
Also, `-4lta-2lt2" and, " 0lta^(2)lt8`
`rArr" "-2ltalt4" and, "-2sqrt2ltaltsqrt2`
`rArr" "-2ltalt2sqrt2" ...(iii)"`
From (i), (ii) and (iii), we obrain that `ain(-2, -2+sqrt2)`.