Question

The point(s) which lies inside the region bounded by the curves $y^2=3x$ and $(x-2{)}^2=-4(y-4)$ is/are

• A. $(1,3)$
• B. $(4,1)$
• C. $(6,4)$
• D. $(4,-3)$

Answer 1

Answer: (B), (D)

The correct options are
B $(4,1)$
D $(4,-3)$

For the points to lie inside the region bounded by the curves $y^2=3x$ and $(x-2{)}^2=-4(y-4)$, they should lie inside both the curves.

For point $(1,3)$
$C_1:(3{)}^2-3(1)>0$
$C_2:(1-2{)}^2+4(3-4)<0$

For point $(4,1)$
$C_1:(1{)}^2-3(4)<0$
$C_2:(4-2{)}^2+4(1-4)<0$

For point $(6,4)$
$C_1:(4{)}^2-3(6)<0$
$C_2:(4-2{)}^2+4(6-4)>0$

For point $(4,-3)$
$C_1:(-3{)}^2-3(4)<0$
$C_2:(4-2{)}^2+4(-3-4)<0$

So points $(4,1)$ and $(4,-3)$ lie inside the region bounded by the curves