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 Pand 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  Y S = 5  units ( x 5 ) 2 + ( y 3 ) 2 = 5 x 2 10 x + 25 + y 2 6 y + 9 = 25 x 2 + y 2 10 x 6 y + 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, Sis 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    O P Q = 1 2 | 0      8     2    0    1      7   0 0 | = 1 2 | 0 + ( 8 ) ( 7 ) + 0 0 ( 1 ) ( 2 ) 0 | = 1 2 | 58 | = 29  units 2

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