Coordinate Geometry Long Question (Question 6)


Question 6:
Solutions by scale drawing will not be accepted.
Diagram below shows a triangle OPQ. Point S lies on the line PQ.

(a) A point Y moves such that its distance from point S is always 5 uints.
      Find the equation of the locus of Y.  
(b) It is given that point P and point Q lie on the locus of Y       .
      Calculate
      (i) the value of k,
      (ii) the coordinates of Q.
(c) Hence, find the area, in uint2, of triangle OPQ.

Solution:
(a)
The equation of the locus Y (x,y) is given by YS=5 units ( x5 ) 2 + ( y3 ) 2 =5 x 2 10x+25+ y 2 6y+9=25 x 2 + y 2 10x6y+9=0
(b)(i)
Given P (2, k) lies on the locus of Y.
(2)2 + (k)2 – 10(2) – 6(k) + 9 = 0  
4 + k2 – 20 – 6k + 9 = 0
k2 – 6k – 7 = 0
(k – 7) (k + 1) = 0
k = 7   or   k = – 1
Based on the diagram, k = 7. 

(b)(ii) 
As P and Q lie on the locus of Y, S is the midpoint of PQ. P = (2, 7), S = (5, 3).
Let the coordinates of Q = (x, y),
( 2+x 2 , 7+y 2 )=( 5,3 ) 2+x 2 =5       and        7+y 2 =3 2+x=10         and       7+y=6 x=8                and       y=1
Coordinates of point Q = (8, –1).
(c)
Area of  OPQ = 1 2 | 0     8     2    0  1     7   0 0 | = 1 2 |0+( 8 )( 7 )+00( 1 )( 2 )0| = 1 2 | 58| =29  units 2