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7.8.4 Coordinate Geometry Long Question (Question 7 & 8)


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 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  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, 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 )+0−0−( −1 )( 2 )−0| = 1 2 | 58| =29  units 2


Question 8:
Diagram below shows a quadrilateral ABCD. Point C lies on the y-axis.

The equation of a straight line AD is 2y = 5x – 21
(a) Find
(i) the equation of the straight line AB,
(ii) the coordinates of A,

(b) A point P moves such that its distance from point D is always 5 units.
Find the equation of the locus of P.


Solution:
(a)(i)
2y=5x−21 y= 5 2 x− 21 2 m AD = 5 2 m AB × m AD =−1 m AB × 5 2 =−1 m AB =− 2 5 Equation of AB y− y 1 = m AB ( x− x 1 ) y+1=− 2 5 ( x+2 ) 5y+5=−2x−4 5y=−2x−9


(a)(ii)
2y=5x−21 ………. ( 1 ) 5y=−2x−9 ………. ( 2 ) ( 1 )×5:10y=25x−105 ………. ( 3 ) ( 2 )×2:10y=−4x−18 ………. ( 4 ) ( 2 )−( 4 ):0=29x−87 x=3 From ( 1 ), 2y=15−21 2y=−6 y=−3 A=( 3 , −3 )

(b)
y=2, 4=5x−21 5x=25 x=5 Point D=( 5, 2 ) PD=5 ( x−5 ) 2 + ( y−2 ) 2 =5 ( x−5 ) 2 + ( y−2 ) 2 =25 x 2 −10x+25+( y 2 −4y+4 )=25 x 2 + y 2 −10x−4y+4=0

2 thoughts on “7.8.4 Coordinate Geometry Long Question (Question 7 & 8)”

    • Hi Tye,
      It is given in the question that point P and point Q lie on the locus of Y.
      PSQ is a straight line and SP = SQ = 5 units, therefore S is the midpoint of PQ.

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