Final answer:
To find the rectangular equation, the trigonometric parametric equations are manipulated using trigonometric identities to eliminate the parameter and express the relationship between x and y. option a is the correct answer.
Step-by-step explanation:
The student is asking which rectangular equation represents the given parametric equations x = 2sec(t) + 1 and y = 6tan(t). To find the rectangular equation, we should eliminate the parameter t. Since sec(t) = 1/cos(t) and tan(t) = sin(t)/cos(t), we can write these relationships as cos(t) = 1/sec(t) and sin(t) = tan(t) * cos(t). Using the trigonometric identity sin2(t) + cos2(t) = 1, we can express sin(t) and cos(t) in terms of x and y.
The rectangular equation that represents the parametric equations x = 2sec(t) + 1 and y = 6tan(t) is (x-1)²/4 - y²/36 = 1 (option a).
To find the rectangular equation, we can substitute the expressions for x and y into the equation and simplify. Multiplying y by y gives us y², and multiplying x-1 by x-1 gives us (x-1)². The coefficient of x² will be the square of the coefficient of sec(t), which is 2, and the coefficient of y² will be the square of the coefficient of tan(t), which is 6. Therefore, we have (x-1)²/4 - y²/36 = 1.