IV. LOCUS ON THE PARABOLA

1. If is a variable point on the parabola , whose focus is S and vertex is S. M, N are the midpoints of SP, OM respectively.

(i) Show that the locus of M is the parabola , and find the locus of N.

(ii) Determine the coordinate ofs of the vertex and the focal length of each locus. Sketch the origincal parabola and the two loci on the same diagram.

[ans: (i) (ii) ; ; ; ]

2. P is a variable point on the parabola , whose vertex is O. M and N are the feet of the perpendiculars from P on to the axis and directrix, respectively.

R is the midpoint of OP and L is the midpoint of MN. Find the locus of (i) R (ii) L and sketch the loci on the same diagram.

[ans: (i) (ii) ]

3. Show that the equation of the tangent at a variable point on the parabola is.

This tangent meets the tangent at the vertex O in R and the axis in Q.

(i) Find the locus of M, the midpoint of QR.

(ii) If ORPQ is a rectangle, find the locus of P.

What relation is there between the focal lengths of the loci.

[ans: (i) (ii) ]

4. P is the point on the parabola , whose focus is S. SHow that the normal at P has equation .

(i) If this normal meets the axis of the parabola in A, find the locus of M, the midpoint of PA.

(ii) The perpendicular from S to the normal meets it in Q. Show that SQ has equation and hence find the coordinates of Q. Determine the equation of the locus of Q. What relation exists between the locus M and of Q.

[ans: (i) (ii) ; ; locus of M, Q are same]

5. P is the point on the parabola . SHow that the equation of the tangent at P is .

State the coordinates of the focus S of the parabola, and prove that the line through S meeting the tangent at right angles in N, has equation . Hence determine the locus of N as P varies on the curve.

[ans: ; locus of N is y = 0 (i.e. x-axis)]

6. Show that the tangent at on the parabola has equation . This tangents meets the parabola in P and Q. Show that the abscissae , of the points P and Q are the roots of the quadratic equation , and write down the value of .

If M(x, y) is the midpoint of PQ, prove that x = -2t , and find y in terms of t. Hence find the equation of the locus of M as T varies.

[ans: -2t; for M, ; locus is ]

7. In each of the following, find the locus of M(x, y) given:

(i) x = 2(p + q), y = 2pq, pq = -3

(ii) x = p + q, y = pq, p = 3q

(iii) x = p + q, , pq = -1

(iv) x = -2(p + q), , pq = 2

[hints: in (iii), (iv) use the identity ]

[ans: (i) y = -6 (ii) (iii) (iv) ]

8. Find the equation of the tangent at on , and show that the tangent at P and intersect at the point T(p+q, pq).

If the abscissa of P is twice that of Q, find the locus of T, and show that the locus of M, the midpoint of PQ, is the parabola .

[ans: locus of T is ]

9. PQ, a variable chord of the parabola subtends a right angle at the vertex O. If p, q are parameters corresponding to P, Q show that pq = -4.

(i) The tangents at P, Q intersect at T; find the locus of T.

(ii) If M is the midpoint of the chord PQ, determine the locus of M.

[ans (i) y = -4 (ii) ]

10. Show that the chord joining the points and on the parabola has equation , and find the condition that PQ passes through the point (0, -2).

Find the equation of the normal at P, and prove that the point of intersection N of the normal at P, Q has coordinates . Determine the equation of the locus N.

[ans: pq = 2; ; locus of N is ]

11. Write down the eqution of the chord joining If and Qon the parabola , and show that if PQ is a focal chord then pq = -1.

(i) Find the equation of the tangent at P and the coordinates of T, the point of intersection of the tangents at P, Q. Hence determine the equation of the locus of T as P, Q vary.

(ii) Find the equation of the normal at P and the coordinates of N, the point of intersection of the normals at P, Q. Hence determine the equation of the locus N.

[ans: (i) ; locus of N is ]

12. From a point P on the parabola , a tangent is drawn. From the focus S, a perpendicular is drawn to meet the tangent at R. Find

(a) the equation of SR (b) the locus of R.

[ans: (a) (b) y = 0]

13. Show that the locus of the midpoints of chords in the parabola , and which pass through the vertex, is another parabola, .

14. Two points P and Q move on the parabola so that the x-coordinates of P and Q differ by a constant value 2a. What is the locus of M, the midpoint of PQ ?

[ans: ]

15. Prove that the locus of the mid-point M of focal chords in the parabola is the parabola .

16. are points on the parabola with parameter p and 1/p . If the tangents at and intersect at R, prove that the locus of R is the line y = a.

17. At point P on the parabola , a normal PK is drawn. From the vertex O a perpendicular OM is drawn to meet the normal at M. Show that the equation of the locus of M as P varies on the parabola is .

18. The chord PQ in the parabola subtends a right angle at the vertex of the parabola. The normals to the parabola at P and Q meet at R.

(a) Prove that pq = -4, where p and q are the parameters of P and Q.

(b) Show that as P and Q take various positions on the parabola, the locus of R is the parabola .

19. (a) P and Q are points on the parabola and . The tangents at P and Q intersect at R so that the angle PRQ equals 90o . Show that the locus of R is the directrix.

(b) If PQ is the focal chord and the tangents at P and Q meet at R, show that the locus of R is the directrix y = -a.

20. If PQ is a focal chord of the parabola ; QR is the tangent at Q and RP is parallel to the axis of the parabola, prove that the locus of R has as its equation .

21. PQ is a focal chord in the parabola . Normals are drawn at P and Q, at the ends of the focal chord to meet at a point R. Find the locus of R for varying focal chords PQ. [ans: ]

22. Tangents are drawn to a parabola from an external point A(), touching the parabola at P and Q.

(a) Prove that the midpoint M of PQ is the point ().

(b) If A moves along the straight line y = x-1 find the equation of the locus of M.

[ans: ]

23. P is a variable point on the parabola , with focus S. SL is drawn perpendicular to the tangent at P, meeting it at L. SM is drawn perpendicular to the normal at P, meeting it at M.

(a) Find the equation of the locus of L

(b) Find the equation of the locus of M

[ans: (a) [ans: y = 0 (b) ].

24. If PN is a normal to the parabola at a variable point P and SN is drawn through the focus S parallel to the tangent at P to cut the normal at N. Prove that the locus of N is

25. P is a point on the parabola with vertex at the origin. A straightline through O parallel to the tangent at P intersects the parabola again at Q. If the tangents at P and Q meet at T, show that the locus of T as P moves on the parabola is

26. PQ is a focal chord in the parabola .

(a) PT is drawn parallel to the tangent at Q at QT is drawn parallel to the tangent at P. Show that the locus of T is

(b) If M is the midpoint of the focal chord PQ and a line through M, parallel to the axis of the parabola, meets the normal at P in A, find the locus of A.

[ans: (b) ]