1. Show that the equation of the tangent at on the parabola (whose vertex 0) is . This tangent meets the x-axis at Z and the y-axis at T. PN is drawn perpendicular to the axis of the parabola meeting it at N, and PM meets the directrix at right angles in M.

Find the coordinates of Z, T, N and prove that

(a) Z is the midpoint of PT, O bisects TN.

(b) units and hence that ZT : ZS = OZ : OS, where S is the focus [hints: show that ]

(c) units and hence that

(d) the tangent PT makes equal angles with the axis of the parabola and with PS [hints: show that PST is isosceles triangle]

(e) the lines SM, TP intersect at right angles at Z.

[ans: Z(ap, 0); T(0, -]; N(0, ]

2. P is the point on the parabola . Prove that the equation of the normal at P is . This normal meets the axis of the parabola in G and the latus rectum in H. N is the foot of the perpendicular from P on the axis.

(a) Find the coordinates of G and show that NG = 2a units.

(b) Show that H has coordinates [, a] and hence prove that SH : SP = NP : NG, where S is the focus.

[ans (i) G(0,+2a)]

3. Write down the equation of the tangent at the point on the parabola , and hence determine the equation of the normal at

(i) the tangent and normal meet the axis of the parabola in A, B, respectively. Find the coordinates of A and B, and show that the focus S is the midpoint AB.

(iii) If the normal at T intersects the x-axis in C, find C and show that . Hence prove that TC : TA = t : 2.

[ans: (i) A(0,); B(0, +2a) (ii) C()]

4. Shpw that the normal at has gradient -1/p and determine its equation.

A line is drawn from the focus S perpendicular to the normal meeting it at Q; show that the equation SQ is px - y = -a . Prove that the coordinates of Q are and hence show that:

(i) Q is the midpoint of PG, where G is the point of intersection of the normal and the axis of the parabola.

(ii) QR is parallel to the axis, wher R is the point of intersection of the tangent at P and the x-axis.

5. O is the vertex and P is any point on the parabola . The line through the midpoint A of OP, parallel to the axis of the parabola meets the x-axis in B. Prove that:

(i) BP is the tangent to the parabola at P.

(ii) M, the midpoint of AB, lies on the parabola.

(iii) the equation of the line from the focus S at right angles to the tangent at P is

x + py = ap, and hence deduce that the perpendicular and the tangent meet at B.

6. Write down the equations of the tangent and normal at the point on the parabola .

(i) If S is the focus, show that the perpendicular distance SN from S to the normal is

(ii) If SM is drawn perpendicular to the tangent meeting it at M, determine the ratio SN : SM.

[ans (ii) ratio t:1]

7. P, Q are points , Q on the parabola . Find the gradient of the chord PQ and prove that the equation of PQ is .

(i) Find the coordinates of the point B where the chord meets the axis of the parabola, and deduce that if OM, ON are the ordinates of P and Q (where O is the vertex) then .

(ii) E, F are the feet of the perpendiculars from P and Q to the directrix. Find the coordinates of G the midpoint of EF, and show that GS has gradient -2/(p + q), where S is the focus. Hence deduce that GS is at right angles to PQ.

[ans: (i) B(0, -apq) (ii) G[a(p+q), -a]]

8. Find the equation of the secant through and Q on the parabola ., and hence deduce the equations of the tangents at P, Q.

The tangents at P and Q intersect in T, and the secant PQ meets the axis of the parabola in R. Show that T has coordinates [a(p+q) , apq] and deduce that the coordinates of R, T equal in magnitude but opposite in sign.

9. Write down the equation of the chord through and Q on the parabola , whose focus is S. Show that the condition for PQ to pass through the focus is that pq = -1, and prove, if PQ is a focal chord, then:

(i) the tangents at P, Q meet at right angles on the directrix (at R, say).

(ii) the tangents and normals at P, Q form a rectangle.

(iii) the length of PQ = . [hints: show that and similarly for SQ].

10. PQ is a chord of the parabola , subtending a right angle at the vertex O, i.e. QOP = 90o. If and Q , show that the gradient of OP is p/2 and hence prove that pq = -4. Deduce that the tangents at P, Q meet on the fixed line y = -4a and that the chord PQ passes through a fixed point on the axis of the parabola. State the coordinates of this point.

[ans: fixed point is (0, 4a)]

11. Find the equation of the line through and the vertex O of the parabola . This line meets the directrix in R. If S is the focus, prove that RS is parallel to the tangent at P.

Write down the equation of the chord PQ in terms of p, q (the parameters corresponding to the points P, Q) and deduce the condition for PQ to pass through S. Show that RQ is parallel to the axis of the parabola.

[ans: PO is px - 2y = 0]

12. Write down the equation of the chord of contact of tangents drawn from the point to the parabola .

(i) If T lies on the directrix, show that this chord of contact passes through the focus S.

(ii) Tangents are drawn from A(0, -2a) to touch in P and Q; show that the equation of the chord of contact PQ is y = 2a, and find its length.

[ans: units]