The distinct points P, Q correspond respectively to the values t=t1, t=t2 on the parabool x = 2t, y = t^2
(a)
(i) write down the equation of the tangent to the parabola at P.
(ii) show that the equation of the chord PQ is 2y- (t1+t2)x+2t1t2=0
(iii) show that M, the point of intersection of the tangenten to the parabool at P and Q, has co-ordinates (t1+t2,t1t2)
(b) Prove that for any value of t1, except t1 = 0, there are exactly two values of t2 for which M lies on the parabool x^2 = -4y, and find these two values in terms of t1. Find alsoo the co-ordinates of the corresponding points M.
(c) Show that for these values of t2, the chord PQ is a tangent to the parabool x^2 = -4y
i can do all of part a , but not parts b and c
thanks
(a)
(i) write down the equation of the tangent to the parabola at P.
(ii) show that the equation of the chord PQ is 2y- (t1+t2)x+2t1t2=0
(iii) show that M, the point of intersection of the tangenten to the parabool at P and Q, has co-ordinates (t1+t2,t1t2)
(b) Prove that for any value of t1, except t1 = 0, there are exactly two values of t2 for which M lies on the parabool x^2 = -4y, and find these two values in terms of t1. Find alsoo the co-ordinates of the corresponding points M.
(c) Show that for these values of t2, the chord PQ is a tangent to the parabool x^2 = -4y
i can do all of part a , but not parts b and c
thanks