could anyone help me to solve this problem
Given a parametrization of the tangent line to the curve,(x(t),y(t))
at t=a is:
x(a)+tx'(a)
y(a)+ty'(a)
Also given that the equation of the tangent line at x=a to y=f(x) is
y=f(a)+f'(a)(x-a)
How does this equation relate to the parametric equation above.
I believe that i muse substitute something from this equation to the one above but i dont know how to do so. And i tried drawing a graph f(x) with the equation of a tangent line to the equation on the point (x(t),y(t)). Could anyone help me with this.
The starting point would be to recognize that: $f'(a) = \dfrac{dy}{dx}|_{x=a} = \dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}_{t=a} = \dfrac{y'(a)}{x'(a)}$. Then you can rewrite the tangent equation as: $\dfrac{y-f(a)}{x-a} = f'(a) = \dfrac{y'(a)}{x'(a)}$, and this gives: $\dfrac{y-f(a)}{y'(a)} = \dfrac{x-a}{x'(a)} = t$. From this you get the two equations in $t$. This shows you can deduce the standard equation of the tangent line from the parametric equations.