Find the Domain p^2+2q^2=11

Problem

p2+2*q2=11

Solution

1. Identify the type of equation. This is the equation of an ellipse in the p*qplane, where p and q are real variables.

2. Isolate the variable p to find its possible range of values. Subtract 2*q2 from both sides.

p2=11−2*q2

3. Apply the property of real squares. Since 2*q2≥0 for all real q the expression 11−2*q2 reaches its maximum value when q=0

p2≤11

4. Solve the inequality for p by taking the square root of both sides.

|p|≤√(,11)

5. Express the domain of p in interval notation.

−√(,11)≤p≤√(,11)

6. Repeat the process for q if the domain of q is required. Subtract p2 from both sides and divide by 2.

q2=(11−p2)/2

7. Solve the inequality q2≤11/2 for q

−√(,11/2)≤q≤√(,11/2)

Final Answer

Domain of *p:[−√(,11),√(,11)], Domain of *q:[−√(,5.5),√(,5.5)]

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