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Let's solve the problem step by step based on the given equation:

### Given:
\[ a = \sqrt{\frac{3b + c}{b - c}} \]
and \( 3b + c > 0 \).

---

### (i) Express \( b \) in terms of \( a \) and \( c \):

We begin by squaring both sides to remove the square root:
\[
a^2 = \frac{3b + c}{b - c}
\]

Now, cross-multiply:
\[
a^2(b - c) = 3b + c
\]

Expand the left side:
\[
a^2b - a^2c = 3b + c
\]

Now, collect terms involving \( b \) on one side:
\[
a^2b - 3b = a^2c + c
\]

Factor out \( b \) on the left side:
\[
b(a^2 - 3) = a^2c + c
\]

Finally, solve for \( b \):
\[
b = \frac{a^2c + c}{a^2 - 3}
\]

---

### (ii) Find the value of \( b \) when \( a = 2 \) and \( c = 5 \):

Substitute \( a = 2 \) and \( c = 5 \) into the expression for \( b \):
\[
b = \frac{2^2 \cdot 5 + 5}{2^2 - 3}
\]

Simplify the terms:
\[
b = \frac{4 \cdot 5 + 5}{4 - 3} = \frac{20 + 5}{1} = 25
\]

Thus, \( b = 25 \).

---

### (iii) For \( b \) to be defined, \( a^2 \neq t \). State the value of \( t \).

From the expression for \( b \), we see that the denominator is \( a^2 - 3 \). For \( b \) to be defined, the denominator cannot be zero, so:
\[
a^2 - 3 \neq 0
\]

This implies:
\[
a^2 \neq 3
\]

Thus, the value of \( t \) is \( 3 \).

---

Would you like any further details or explanations?

Here are five related questions to extend your understanding:
1. How would the result change if \( a = 3 \) and \( c = 2 \)?
2. Can you solve the equation for \( c \) in terms of \( a \) and \( b \)?
3. What conditions must be satisfied for \( a \) to be a real number?
4. How would the expression for \( b \) change if the equation was \( a = \sqrt{\frac{4b + c}{b - c}} \)?
5. What happens if \( a^2 = 3 \), and why does that make \( b \) undefined?

**Tip**: Always check that the domain of your variables ensures that denominators do not become zero, as this makes the expression undefined.

Given that a = √((3b + c) / (b - c)), and 3b + c > 0, express b in terms of a and c, find the value of b when a = 2 and c = 5, and for b to be defined, a^2 ≠ t. State the value of t.

Learn how to express b in terms of a and c using algebraic manipulation. Solve for b when a = 2 and c = 5, and determine the conditions for b to be defined in the equation a = √((3b + c) / (b - c)). Ideal for students in Grades 11-12.

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