General And Middle Terms

Step 1: Understanding the Concept:
General term: .

Step 2: Detailed Explanation:
Power of x: . So constant term = .

Step 3: Final Answer:
Option (E).

Concept: From identities:

Step 1: Interpret given sums.


Step 2: Use identity.


Step 3: Find required value.

Concept:

Step 1: Form ratio.


Step 2: Add 1.

Concept: General term:

Step 1: Coefficient of .
Coefficient:

Step 2: Coefficient of .
Coefficient:

Step 3: Equate coefficients.


Step 4: Simplify.

Concept:
• General term of binomial expansion:
• Match power of to required term

Step 1: Write general term


Step 2: Simplify powers of


Step 3: Find power condition


Step 4: Substitute
Final Conclusion:
Coefficient =

Step 1: Complete the square.
We begin by completing the square for the given equation of the circle: First, group the -terms and the -terms: Now, complete the square for both the -terms and the -terms:
- For , add and subtract .
- For , add and subtract . The equation becomes:
Step 2: Relate to the area of the circle.
The standard equation for a circle is: Here, , so the area of the circle is , which becomes: