If X is between – 3 and – 1, and Y is between – 1 and 1, then $X^2-Y^2$ is in between which of the following?

The correct option is 0 and 9.

To find the range of values for the expression
$X^2-Y^2$
we need to determine the range of both
$X^2$
and
$Y^2$
individually, and then analyze their difference.

Step 1: Determine the range of
$X^2$
We are given that
$X$
is strictly between -3 and -1:
$-3<X<-1$
Since both bounds are negative, squaring the terms reverses the direction of the inequality signs:
${(-3)}^2>X^2>{(-1)}^2$
$9>X^2>1$
Writing this in standard order gives:
$1<X^2<9$

Step 2: Determine the range of
$Y^2$
We are given that
$Y$
is strictly between -1 and 1:
$-1<Y<1$
Since this interval contains 0, the minimum possible value of
$Y^2$
is 0 (which occurs exactly when
$Y=0$
). The maximum value of
$Y^2$
approaches 1 but is strictly less than 1. Thus, the range is:
$0\le Y^2<1$

Step 3: Find the range of
$X^2-Y^2$
To find the lower and upper bounds of the difference, we evaluate the extreme scenarios:
• Lower Bound: This is achieved by subtracting the largest possible value of
$Y^2$
from the smallest possible value of
$X^2$
.
Since
$X^2>1$
and
$Y^2<1$
, the lower bound is:
$1-1=0$
• Upper Bound: This is achieved by subtracting the smallest possible value of
$Y^2$
from the largest possible value of
$X^2$
.
Since
$X^2<9$
and
$Y^2\ge 0$
, the upper bound is:
$9-0=9$

Combining these bounds, we find:
$0<X^2-Y^2<9$
Thus, the expression is strictly between 0 and 9.