Just differentiate x and y separately with respect to t.

Then take dy/dt divide by dx/dt to get dy/dx.


Then for tangent is horizontal means dy/dx is zero

To calculate this equate dy/dx to zero and solve for t.
Then sub the value of t found into the equation for x and y in terms of t given in question to get the x and y coordinates of the point

For equation of tangent where t=2, sub t=2 into dy/dx and x and y to get gradient x and y coordinates then use Y-y=m(X-x) to get equation of tangent l

To determine whether l meets C again, sub the equation of x and y in terms of t into the equation of tangent solved. Then solve for value(s) of t.
If more than 1 value the tangent meets C again.
If only 1 value of t (which should be t=2) then the tangent does not meet the curve again