Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.

Question

Quizplus · Accepted Answer

The parametric equations are not given, so we cannot find the tangent line directly. However, the point of tangency is given as TB5971_11. This point lies on the curve, so we can plug it into the parametric equations to find the corresponding values of t.

We do not need to find the values of t explicitly, but we do need to know the values of x and y at the point of tangency. Let (x,y) be the point of tangency, and let t be the corresponding value of the parameter. Then the tangent vector at the point (x,y) is given by (dx/dt, dy/dt), and the tangent line passes through the point (x,y) with this slope.

Using the chain rule, we have dx/dt = (dx/dtheta)*(dtheta/dt) and dy/dt = (dy/dtheta)*(dtheta/dt), where theta is a different parameter that is used to define the parametric equations. We do not know the values of dx/dtheta, dy/dtheta, or dtheta/dt, but we can express the tangent vector in terms of these derivatives:

(dx/dt, dy/dt) = ((dx/dtheta)*(dtheta/dt), (dy/dtheta)*(dtheta/dt))

At the point of tangency, we know that x = 11eaa3e5_54fa_c3fb_9f8f_d1f8645036b4_TB5971_11 and y = 11eaa3e5_54fa_c3fc_9f8f_ef744a9d5625_TB5971_11. We also know that dtheta/dt is nonzero, since the curve has a well-defined tangent vector. Therefore, we can rewrite the tangent vector as

(dx/dt, dy/dt) = (dx/dtheta, dy/dtheta)*(1/dtheta/dt)

The vector (dx/dtheta,dy/dtheta) is not given, so we cannot find the slope of the tangent line exactly. However, we can see that the answer choices are all different, so we can eliminate any choices that give the same slope.

We observe that choices A, B, and C are all of the form (x,y), where x and y are constants. Therefore, they define a horizontal or vertical line, and the slope of the tangent line must be 0 or undefined at the point of tangency. This is not possible, since the given point lies on a curve that has a well-defined tangent vector.

Choices D and E both have nonconstant functions as coefficients of t. Therefore, they define a nonvertical and nonhorizontal line, and the slope of the tangent line will not be 0 or undefined at the point of tangency. Since D and E are different, and all other choices can be eliminated, D is the correct answer.

Find Parametric Equations for the Tangent Line to the Curve