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SPM Additional Mathematics 2019, Paper 2 (Question 9)


Question 9:
Solution by scale drawing is not accepted.
Diagram 5 shows the path of a moving point P(x, y). P always moves at a constant distance from point A.
Diagram 5

B(–1, –2) and R(–5, q) lie on path of point P. The straight line BC is a tangent to the path and
intersects the x-axis at point C.
Find
(a) the equation of the path of point P, [3 marks]
(b) the possible values of q, [2 marks]
(c) the area of ∆ ABC. [5 marks]


Solution:
(a)
Radius AB=(2+1)2+(1+2)2                =1+9                =10PA=10(x+2)2+(y1)2=10(x+2)2+(y1)2=10x2+4x+4+y22y+110=0x2+y2+4x2y5=0

(b)
When x = –5, y = q,
(–5)2 + q2 + 4(–5) – 2q – 5 = 0
25 + q2 – 20 – 2q – 5 = 0
q2 – 2q = 0
q(q – 2) = 0
q = 0, q = 2

(c)
mAB×mBC=1(1+22+1)×mBC=1(31)×mBC=1mBC=13Equation of BC:y(2)=13(x+1)3(y+2)=x+13y+6=x+13y=x5At point C, y=0x5=0x=5C=(5, 0)

Area of ΔABC=12| 5   2   1     5  0      1    2    0|=12|5+4+00(1)(10)|=12|20|=10 unit2

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