1. Locus Problems

2. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)

3. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)
We find the locus of a point.

4. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)
We find the locus of a point.
1 Find the coordinates of the point whose locus you are finding.

5. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)
We find the locus of a point.
1 Find the coordinates of the point whose locus you are finding.
2 Look for the relationship between the x and y values
(i.e. get rid of the parameters)

6. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)
We find the locus of a point.
1 Find the coordinates of the point whose locus you are finding.
2 Look for the relationship between the x and y values
(i.e. get rid of the parameters)
a) Type 1: already no parameter in the x or y value.

7. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)
We find the locus of a point.
1 Find the coordinates of the point whose locus you are finding.
2 Look for the relationship between the x and y values
(i.e. get rid of the parameters)
a) Type 1: already no parameter in the x or y value.
e.g. (p + q,– 3)

8. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)
We find the locus of a point.
1 Find the coordinates of the point whose locus you are finding.
2 Look for the relationship between the x and y values
(i.e. get rid of the parameters)
a) Type 1: already no parameter in the x or y value.
e.g. (p + q,– 3)
locus is y = – 3

9. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)
We find the locus of a point.
1 Find the coordinates of the point whose locus you are finding.
2 Look for the relationship between the x and y values
(i.e. get rid of the parameters)
a) Type 1: already no parameter in the x or y value.
e.g. (p + q,– 3)
locus is y = – 3
b) Type 2: obvious relationship between the values, usually only one
parameter.

10. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)
We find the locus of a point.
1 Find the coordinates of the point whose locus you are finding.
2 Look for the relationship between the x and y values
(i.e. get rid of the parameters)
a) Type 1: already no parameter in the x or y value.
e.g. (p + q,– 3)
locus is y = – 3
b) Type 2: obvious relationship between the values, usually only one
parameter.
e.g. 6t , t 2  1

11. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)
We find the locus of a point.
1 Find the coordinates of the point whose locus you are finding.
2 Look for the relationship between the x and y values
(i.e. get rid of the parameters)
a) Type 1: already no parameter in the x or y value.
e.g. (p + q,– 3)
locus is y = – 3
b) Type 2: obvious relationship between the values, usually only one
parameter.
e.g. 6t , t 2  1 x  6t
x
t
6

12. Locus Problems
Eliminate the parameter (get rid of the p’s and q’s)
We find the locus of a point.
1 Find the coordinates of the point whose locus you are finding.
2 Look for the relationship between the x and y values
(i.e. get rid of the parameters)
a) Type 1: already no parameter in the x or y value.
e.g. (p + q,– 3)
locus is y = – 3
b) Type 2: obvious relationship between the values, usually only one
parameter.
e.g. 6t , t 2  1 x  6t y  t 2 1
x x2
t y  1
6 36

13. c) Type 3: not an obvious relationship between the values, use a
previously proven relationship between the parameters.

14. c) Type 3: not an obvious relationship between the values, use a
previously proven relationship between the parameters.
e.g.  p 2  q 2 , p  q 
Given that pq = 3

15. c) Type 3: not an obvious relationship between the values, use a
previously proven relationship between the parameters.
e.g.  p  q , p  q 
2 2
x  p2  q2
  p  q   2 pq
2
Given that pq = 3

16. c) Type 3: not an obvious relationship between the values, use a
previously proven relationship between the parameters.
e.g.  p  q , p  q 
2 2
x  p2  q2
x  y2  6
  p  q   2 pq
2
Given that pq = 3

17. c) Type 3: not an obvious relationship between the values, use a
previously proven relationship between the parameters.
e.g.  p  q , p  q 
2 2
x  p2  q2
x  y2  6
  p  q   2 pq
2
Given that pq = 3
2005 Extension 1 HSC Q4c)
The points P2ap, ap 2  and Q2aq, aq 2  lie on the parabola x 2  4ay
The equation of the normal to the parabola at P is x  py  2ap  ap 3
and the equation of the normal at Q is given by x  qy  2aq  aq 3

18. c) Type 3: not an obvious relationship between the values, use a
previously proven relationship between the parameters.
e.g.  p  q , p  q 
2 2
x  p2  q2
x  y2  6
  p  q   2 pq
2
Given that pq = 3
2005 Extension 1 HSC Q4c)
The points P2ap, ap 2  and Q2aq, aq 2  lie on the parabola x 2  4ay
The equation of the normal to the parabola at P is x  py  2ap  ap 3
and the equation of the normal at Q is given by x  qy  2aq  aq 3
(i) Show that the normals at P and Q intersect at the point R whose
coordinates are  apq p  q , a p 2  pq  q 2  2

19. c) Type 3: not an obvious relationship between the values, use a
previously proven relationship between the parameters.
e.g.  p  q , p  q 
2 2
x  p2  q2
x  y2  6
  p  q   2 pq
2
Given that pq = 3
2005 Extension 1 HSC Q4c)
The points P2ap, ap 2  and Q2aq, aq 2  lie on the parabola x 2  4ay
The equation of the normal to the parabola at P is x  py  2ap  ap 3
and the equation of the normal at Q is given by x  qy  2aq  aq 3
(i) Show that the normals at P and Q intersect at the point R whose
coordinates are  apq p  q , a p 2  pq  q 2  2
1
(ii) The equation of the chord PQ is y   p  q x  apq . If PQ passes
through (0,a), show that pq = – 1 2

20. (iii) Find the locus of R if PQ passes through (0,a)

21. (iii) Find the locus of R if PQ passes through (0,a)
x  apq p  q 

22. (iii) Find the locus of R if PQ passes through (0,a)
x  apq p  q 
x  a p  q 
x
pq 
a

23. (iii) Find the locus of R if PQ passes through (0,a)
x  apq p  q 
x  a p  q 
x
pq 
a
y  a p 2  pq  q 2  2

24. (iii) Find the locus of R if PQ passes through (0,a)
x  apq p  q 
x  a p  q 
x
pq 
a
y  a p 2  pq  q 2  2

y  a  p  q   pq  2
2


25. (iii) Find the locus of R if PQ passes through (0,a)
x  apq p  q 
x  a p  q 
x
pq 
a
y  a p 2  pq  q 2  2

y  a  p  q   pq  2
2

 x2 
y  a 2  1  2 
a 

26. (iii) Find the locus of R if PQ passes through (0,a)
x  apq p  q 
x  a p  q 
x
pq 
a
y  a p 2  pq  q 2  2

y  a  p  q   pq  2
2

 x2 
y  a 2  1  2 
a 
x2
y   3a
a

27. 2004 Extension 1 HSC Q4b)
The two points P2ap, ap 2  and Q2aq, aq 2  are on the parabola x 2  4ay

28. 2004 Extension 1 HSC Q4b)
The two points P2ap, ap 2  and Q2aq, aq 2  are on the parabola x 2  4ay
(i) The equation of the tangent to x 2  4ay at an arbitrary point 2at , at 2 
on the parabola is y  tx  at
2
Show that the tangents at the points P and Q meet at R, where R is
the point a p  q , apq

29. 2004 Extension 1 HSC Q4b)
The two points P2ap, ap 2  and Q2aq, aq 2  are on the parabola x 2  4ay
(i) The equation of the tangent to x 2  4ay at an arbitrary point 2at , at 2 
on the parabola is y  tx  at
2
Show that the tangents at the points P and Q meet at R, where R is
the point a p  q , apq
(ii) As P varies, the point Q is chosen so that POQ is a right angle,
where O is the origin.
Find the locus of R.

30. 2004 Extension 1 HSC Q4b)
The two points P2ap, ap 2  and Q2aq, aq 2  are on the parabola x 2  4ay
(i) The equation of the tangent to x 2  4ay at an arbitrary point 2at , at 2 
on the parabola is y  tx  at
2
Show that the tangents at the points P and Q meet at R, where R is
the point a p  q , apq
(ii) As P varies, the point Q is chosen so that POQ is a right angle,
where O is the origin.
Find the locus of R.
mOP  mOQ  1

31. 2004 Extension 1 HSC Q4b)
The two points P2ap, ap 2  and Q2aq, aq 2  are on the parabola x 2  4ay
(i) The equation of the tangent to x 2  4ay at an arbitrary point 2at , at 2 
on the parabola is y  tx  at
2
Show that the tangents at the points P and Q meet at R, where R is
the point a p  q , apq
(ii) As P varies, the point Q is chosen so that POQ is a right angle,
where O is the origin.
Find the locus of R.
mOP  mOQ  1
ap 2  0 aq 2  0
  1
2ap  0 2aq  0

32. 2004 Extension 1 HSC Q4b)
The two points P2ap, ap 2  and Q2aq, aq 2  are on the parabola x 2  4ay
(i) The equation of the tangent to x 2  4ay at an arbitrary point 2at , at 2 
on the parabola is y  tx  at
2
Show that the tangents at the points P and Q meet at R, where R is
the point a p  q , apq
(ii) As P varies, the point Q is chosen so that POQ is a right angle,
where O is the origin.
Find the locus of R.
mOP  mOQ  1
ap 2  0 aq 2  0
  1
2ap  0 2aq  0
ap 2 q 2  4a 2 pq

33. 2004 Extension 1 HSC Q4b)
The two points P2ap, ap 2  and Q2aq, aq 2  are on the parabola x 2  4ay
(i) The equation of the tangent to x 2  4ay at an arbitrary point 2at , at 2 
on the parabola is y  tx  at
2
Show that the tangents at the points P and Q meet at R, where R is
the point a p  q , apq
(ii) As P varies, the point Q is chosen so that POQ is a right angle,
where O is the origin.
Find the locus of R.
mOP  mOQ  1
ap 2  0 aq 2  0
  1
2ap  0 2aq  0
ap 2 q 2  4a 2 pq
pq  4

34. 2004 Extension 1 HSC Q4b)
The two points P2ap, ap 2  and Q2aq, aq 2  are on the parabola x 2  4ay
(i) The equation of the tangent to x 2  4ay at an arbitrary point 2at , at 2 
on the parabola is y  tx  at
2
Show that the tangents at the points P and Q meet at R, where R is
the point a p  q , apq
(ii) As P varies, the point Q is chosen so that POQ is a right angle,
where O is the origin.
Find the locus of R.
mOP  mOQ  1
ap 2  0 aq 2  0 y  apq
  1
2ap  0 2aq  0
ap 2 q 2  4a 2 pq
pq  4

35. 2004 Extension 1 HSC Q4b)
The two points P2ap, ap 2  and Q2aq, aq 2  are on the parabola x 2  4ay
(i) The equation of the tangent to x 2  4ay at an arbitrary point 2at , at 2 
on the parabola is y  tx  at
2
Show that the tangents at the points P and Q meet at R, where R is
the point a p  q , apq
(ii) As P varies, the point Q is chosen so that POQ is a right angle,
where O is the origin.
Find the locus of R.
mOP  mOQ  1
ap 2  0 aq 2  0 y  apq
  1
2ap  0 2aq  0 y  4a
ap 2 q 2  4a 2 pq
pq  4

36. 2004 Extension 1 HSC Q4b)
The two points P2ap, ap 2  and Q2aq, aq 2  are on the parabola x 2  4ay
(i) The equation of the tangent to x 2  4ay at an arbitrary point 2at , at 2 
on the parabola is y  tx  at
2
Show that the tangents at the points P and Q meet at R, where R is
the point a p  q , apq
(ii) As P varies, the point Q is chosen so that POQ is a right angle,
where O is the origin.
Find the locus of R.
mOP  mOQ  1
ap 2  0 aq 2  0 y  apq
  1 Exercise 9J; odds
2ap  0 2aq  0 y  4a up to 23
ap 2 q 2  4a 2 pq
pq  4