Long Questions (Question 6)


Question 7:
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 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=5x210x+25+y26y+9=25x2+y210x6y+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, is the midpoint of PQ. P = (2, 7), S = (5, 3).
Let the coordinates of Q = (x, y),
(2+x2,7+y2)=(5,3)2+x2=5   and    7+y2=32+x=10 and    7+y=6x=8and    y=1
Coordinates of point Q = (8, –1).

(c)
Area of  OPQ=12|0 8  2   0  1 7  00|=12|0+(8)(7)+00(1)(2)0|=12|58|=29 units2