Find the Domain ((x-3)^2)/81-(y^2)/144=1

Problem

((x−3)2)/81−(y2)/144=1

Solution

1. Identify the type of equation. This is the standard form of a horizontal hyperbola, which is defined by the general equation ((x−h)2)/(a2)−((y−k)2)/(b2)=1

2. Determine the orientation. Since the x term is positive and the y term is negative, the hyperbola opens to the left and to the right.

3. Find the center and the horizontal distance to the vertices. The center (h,k) is (3,0) and a2=81 which means a=9

4. Establish the inequality for the domain. For a horizontal hyperbola, the expression ((x−h)2)/(a2) must be greater than or equal to 1 to ensure that y has real values.

((x−3)2)/81≥1

5. Solve the inequality for x Multiply both sides by 81 and take the square root.

(x−3)2≥81

|x−3|≥9

6. Split the absolute value into two cases.

x−3≤−9* or *x−3≥9

x≤−6* or *x≥12

Final Answer

Domain:(−∞,−6]∪[12,∞)

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