Thread: Two Tangents at the same Point?

1. Two Tangents at the same Point?

Problem:
- Find equations of the tangents to the curve and that pass through the point (4,3).

$\displaystyle \frac{dy}{dx}=\frac{6t^2}{6t}=t$

So the deriviative here is t. When I solve the parametric equations at the point (4,3), I get $\displaystyle t=\pm1$ for the quadratic equation, and t=1 for the cubic equation. The only solution common between them is t=1. So how can there be two tangents through this point? Note that the problem explicity states that there are two tangents. I would expect there to be two solutions for t at this point if this were true.

2. The tangent at the point $\displaystyle (3t^2+1,2t^3+1)$ has gradient t, so its equation is $\displaystyle y-2t^3-1 = t(x-3t^2-1)$. The condition for this line to pass through the point (4,3) is obtained by putting y=3 and x=4 in the equation of the tangent. That will give you a cubic equation in t, which has a repeated root t=1 and also another root.