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#### xwrathbringerx

##### Guest

**1.**P(2ap, ap^2) and Q(2aq,aq^2) are 2 points on the parabola x^2 = 4ay. Tangents to the parbaola at P and Q intersect at the point T.

a) Show that the equation of the tangent at P is y=px-ap^2.

b) Find the coordinates of T.

c) P and Q move on the parabola so that the line PQ passes through the point (2a,-a). Show that p + q + 1 = pq.

d) Hence, by finding the Cartesian equation of the locus T, show that T lies on a straight line.

**e) With the aid of a diagram, carefully explain why the locus of T is not all of the straight line.**

Could someone please show me how to do (e). I've found out that the locus of T is x-y+a = 0 and drawn myself a diagram with all the info provided and obtained but I have no clue how to use these to prove (e).

**2.**P(2ap, ap^2), Q(2aq,aq^2) and R(2ar,ar^2) are points on the parabola x^2 = 4ay.

a) Show that the equation of the normal at P is py+x = 2ap + ap^3.

**b) Find the coordinates of the point of intersection of the normals at P and Q.**

Is the pt of intersection (-ap^2q+apq^2 , 2a + a(p^2+pq+q^2)?

**c) If the normals P,Q and R are constant, show that p +q +r=0.**

How on earth do you do this??

Could someone please help me with these questions? Thanxx