Question

Find the equations, in slope - intercept form, of the lines tangent to the parametric curve at the origin, given:

x = t³ - 9t
y = 9t² - t⁴
For the negative value of t , the tangent line is
y =

Answer 1

The equations of the lines tangent to the parametric curve at the origin are y = 0.

Step-by-step explanation:

To find the equations of the lines tangent to the parametric curve at the origin, we first need to find the derivatives of the given parametric equations with respect to t.

The derivatives are:

x' = 3t^2 - 9

y' = 18t - 4t^3

The slope of the tangent line at the origin is given by the value of y' when t = 0.

Substituting t = 0 into y', we get y' = 0.

Therefore, the slope of the tangent line at the origin is 0.

The equation of a line with a slope of 0 is y = b, where b is the y-intercept.

Since the tangent line passes through the origin, the y-intercept is 0.

Therefore, the equation of the tangent line at the origin is y = 0.