If a < x < b then for which of the following relations does 0 < y < 1 always hold?

Correct option is B

Option Analysis:

1. Option a: y = (a - x) / (b + a)

   • For a < x < b, the term (a - x) would be negative because x is greater than a.
   • Therefore, y would be negative, which does not satisfy 0 < y < 1.
   • This option does not work.

2. Option b: y = (x - a) / (b - a)

   • For a < x < b, the term (x - a) is positive and less than (b - a).
   • Dividing by (b - a), which is positive, gives 0 < y < 1.
   • This option satisfies the condition 0 < y < 1.

3. Option c: y = (x - b) / (b - a)

   • For a < x < b, the term (x - b) would be negative because x is less than b.
   • Therefore, y would be negative, which does not satisfy 0 < y < 1.
   • This option does not work.

4. Option d: y = (b - x) / (a + b)

   • For a < x < b, the term (b - x) is positive, but we need to check if this keeps y within 0 < y < 1.
   • However, the denominator (a + b) does not necessarily ensure that y falls between 0 and 1, as it is not directly related to the range (b - a).
   • This option does not consistently satisfy 0 < y < 1.

Conclusion

The only option that ensures 0 < y < 1 for a < x < b is:

Answer: Option b, y = (x - a) / (b - a)