F.4 AM - Ans.fyi 參考答案

F.4 AM

2007-10-01 3:17 pm

Let f(x)=k-(x-h)^2 where h and k are positive constants.
(a)suppose the graph of y=f(x) cuts the x-axis at P and Q. Find.in terms of h and k
i)the length of PQ
ii)the coordinates of the mid-point M of PQ
(bi)If one root of the equation f(x)=0 is twice the other, expree h in term of k.
(bii)If, in addition, the product of the roots is equal to 32, find the values of h and k

回答 (1)

用戶238286

2007-10-01 5:05 pm

✔ 最佳答案

(ai) f(x)=k-(x-h)^2,set f(x)=0,
k-(x-h)^2=0
k-(x^2-2xh+h^2)=0
x^2-2xh+h^2-k=0

sum of roots(a+b):2h,product of roots(ab):h^2-k

(a-b)^2=(a+b)^2-4ab
a-b=square root of(4h^2-4h^2-4k)
a-b=-2square root k
the length of PQ is 2 square root k.

(aii)the coordinates of M is(x,0)
x=(a+b)/2=2h/2=h
so,the coordinates of mid-point is (h,0)

(bi) a,b=2a
a+2a=2h,3a=2h,a=2h/3
b=2(2h/3)=4h/3
ab=h^2-k
(2h/3)(4h/3)=h^2-k
8h^2/9=h^2-k
8h^2=9h^2-9k
h^2=9k
h=3 square root k

(bii)a,b=2a
the product of root:a(2a)=32
2a^2=32
a=4
b=2(4)=8
a+b=2h
12=2h
h=6
h=3 square root k
6=3 square root k
k=4

參考: myself

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