Two Tangent lines

Question

Find the tangent equation to the given curve that passes through the point (5, 5). Note that due to the t²

 in the x equation and the t³  in the y equation, the equation in the parameter t has more than one solution. This means that there is a second tangent equation to the given curve that passes through a different point.

x = 3t² + 2 

y = 2t³ + 3

T.L at smaller t value: ?

T.L at larger t value: ?

Answer 1

x = 3t^2 + 2

t^2 = (x-2)/3

t=sqr[(x-2)/3]^3 = [(x-2)/3]^(3/2)

y = 2t^3 + 3

y= 2[(x-2)/3]^(3/2) + 3

y= (2/sqr27)(x-2)^(3/2) + 3

y'= (2/sqr27)(3/2)(x-2)^(1/2)

y'(5) = (6/2sqr27)sqr3 = 3(sqr(3/27) = 3/sqr9 = 3/3 = 1 = slope of the tangent line

y=x+b

5=5+b

b = 0

y=x is the tangent line equation at the point (5,5) for the parametric equations x=3t^2 +3 and y=2t^3 +2

the calculations were tedious, and very subject to potential error,

but when the answer comes out so simple, it's a clue it might be right

(nonetheless no guarantees it's right. too many potential errors)

-2 seems to be the slope for the 2nd tangent line

y=-2x +b