F4 Maths

42.
a=(x+y-1)/[(x+y)²-1]=(x+y-1)/(x+y+1)(x+y-1)=1/(x+y+1)
b=(x-y+1)/[(x-y)²-1]=(x-y+1)/(x-y+1)(x-y-1)=1/(x-y-1)
(b+1)/(b-1)=[1/(x-y-1)+1]/[1/(x-y-1)-1]=[1+(x-y-1)]/[1-(x-y-1)]
=(x-y)/(2-x+y)
(a-1)/(a+1)=[1/(x+y+1)-1]/[1/(x+y+1)+1]=[1-(x+y+1)]/[1+(x+y+1)]
=(-x-y)/(2+x+y)
(b+1)(a-1)/(b-1)(a+1)=-(x-y)(x+y)/(2-x+y)(2+x+y)

43.
(a)
b=1/x-1/y=(y-x)/xy=-(x-y)/xy
ab=-xy(x-y)/xy(x-y)=-1
(b)
(a²+ab)(ab-b)/(b²+ab)(ab+a)=(a²-1)(-1-b)/(b²-1)(-1-a)
=(a+1)(a-1)(b+1)/(b+1)(b-1)(a+1)=(a-1)/(b-1)
=[xy/(x-y)-1]/[-(x-y)/xy-1]=[xy-(x-y)]/-(x-y)[(x-y)/xy+1]
=xy(xy-x+y)/(y-x)(x-y+xy)

23.
x²+y²=20 and y=mx+6
x²+(mx+6)²=20
(m²+1)x²+12mx+16=0
△=144m²-64(m²+1)=80m²-64>0
5m²-4>0
m<-2/√5 or m>2/√5
Ans:-1

24.
When x=0
y=-c=-2
c=2

a.
3x-x²/8=0
x=24
(24,0)
5y=3(12)-(12)²/8=20,y=4
(12,4)
Ans:44

圖片參考：http://upload.lyfhk.net/di-6NLI.jpg

(on your rough paper)
5y=3x-x²/8
2y+x=0,x=-2y
5y=-6y-y²/2
y²/2+11y=0
y=-22,x=44
(44.-22)

b.
2y+(44)=0,y=-22
√(x²+y²)=22√5=49
Ans:49

17.
p-q+6=0,q=p+6
3p(p/4+1)-q+4=0
3p²/4+3p-p-6+4=0
3p²/4+2p-2=0
3p²+8p-8=0
p=[-8±√(64+96)]/6=(-4±2√10)/3
q=(-4±2√10)/3+6=(14±2√10)/3