大 家 帮 帮 忙 教 教 我 怎 么 做

jernie

大 家 帮 帮 忙  教 教 我 怎 么 做

1. if the line y=3x+c is a tangent to the ellipse x^2+4y^2 = 4 ,find c.

2. find the parametric equation of the ellipse (x+1)^2 + 4 (y-1)^2 = 4

3. AB is a diameter of the ellipse x^2 +9y^2 = 25. The eccentric angle of A is π/6 .Find the eccentric angle of B.

4.find the equation of the locus of midpt of the chords of the hyperbola x^2 -4y^2=4 that hv a gradient of 2.

5.if the roots of the equation x^4-px+qx^2-pqx+1=0 are  α, β, γ,&  δ.find the value of (α+β+ γ) ( α+β+δ) (  α+ γ+δ) (  β+ γ+ δ)

6. if the equation x^3 +2x^2-5x+1=0 has roots α, β, γ find the equation with roots α+ β, β+γ, α+γ

7. determine the value of m such that the equations x^3 + mx -1 = 0 and x^3 - 3x +m = 0 hv a common root

8.if 1-a tanθ=secθ  , b- cotθ  = cosec θ .find the value of a.b

9. evaluate log9 √3 + log 9 √√3 + log 9 √√√3 + ....   (log 9 的 9是 底 数 )

10.if m ( √3/2     - 1/2  )       find m^6
                1/2        √3/2


11. find the coefficient od a^4bc in the expansion of (a+b+c+d)^6

12.if the lines 7x-5y+3 = 0 , 11x+2y+3=0,11x+2y+1=0 and 23x+23y+k=o are concurrent, find k.

13.if w is a complex cube root of unity , evaluate ( 1+ 3w+w^2)^2 + ( 1 + w+3w^2)^2

14.if a = ∫1  1/ 1+x^3 dx , b = ∫1 1/1+x^4dx and c= ∫1 1/1+x^5dx.
                0                                0                                 0
     wat is the relationship between a, b & c ??


希 望 大 家 可 以 帮 到 我
如 果 可 以 的 话 把 做 法 告 诉 我
谢 谢

dunwan2tellu

1. if the line y=3x+c is a tangent to the ellipse x^2+4y^2 = 4 ,find c.

把 y = 3x+c 带入 x^2 + 4y^2 = 4 得到一个 quadratic equation in term of x（里面也有 c).之后用 b^2-4ac = 0 (因为 tangent 的意思是intersect at one point)

2. find the parametric equation of the ellipse (x+1)^2 + 4 (y-1)^2 = 4

考虑 sin^2 t + cos^2 t = 1 的 pattern .
比较原本的 equation : ((x+1)/2)^2 + (y-1)^2 = 1
结论是 sin t = (x+1)/2 , y-1 = cos t
所以 x = ... , y = .... in term of t

3. AB is a diameter of the ellipse x^2 +9y^2 = 25. The eccentric angle of A is π/6 .Find the eccentric angle of B.

4.find the equation of the locus of midpt of the chords of the hyperbola x^2 -4y^2=4 that hv a gradient of 2.

5.if the roots of the equation x^4-px^3+qx^2-pqx+1=0 are  α, β, γ,&  δ.find the value of (α+β+ γ) ( α+β+δ) (  α+ γ+δ) (  β+ γ+ δ)

为了方便，让那四个root 是 a,b,c,d .
sum of root = a+b+c+d = p
所以 (a+b+c)(a+b+d)(b+c+d)(a+c+d)=(p-a)(p-b)(p-c)(p-d)
但是 f(x) =  x^4-px^3+qx^2-pqx+1 = (x-a)(x-b)(x-c)(x-d) (因为a,b,c,d市roots of f(x)) .所以也像等于找 f(p) = (p-a)(p-b)(p-c)(p-d)=p^4-p^4+qp^2-p^2q+1 = 1

6. if the equation x^3 +2x^2-5x+1=0 has roots α, β, γ find the equation with roots α+ β, β+γ, α+γ

a+b+c = -2 , ab+bc+ca = -5 , abc = -1
new equation :
coefficient of x^2 = -[(a+b) + (a+c) + (b+c)] = -[2(a+b+c)] = 4
coefficient of x = (a+b)(a+c) + (a+b)(b+c) + (b+c)(c+a)
=(-2-a)(-2-b) + (-2-a)(-2-c) + (-2-b)(-2-c)
=ab+bc+ca + 4(a+b+c)+12 = -1
coefficent of constant = -(a+b)(a+c)(b+c)=-[(a+b+c)(ab+bc+ca)-abc] = -11
new equation : x^3 + 4x^2 - x - 11 = 0

7. determine the value of m such that the equations x^3 + mx -1 = 0 and x^3 - 3x +m = 0 hv a common root

8.if 1-a tanθ=secθ  , b- cotθ  = cosec θ .find the value of a.b

shortform : sinθ = s , cosθ = c ,所以
1 - as/c = 1/c ==> c - as = 1 ...(1)
b - c/s = 1/s ==> bs - c = 1 ...(2)
(1) : a = (c-1)/s
(2) : b = (c+1)/s
=> a.b = (c^2-1)/s^2 = -s^2/s^2 = -1

9. evaluate log9 √3 + log 9 √√3 + log 9 √√√3 + ....   (log 9 的 9是 底 数 )

注意到 log9 √3 = log9 9^(1/4) = 1/4,log 9 √√3=log9 9^(1/8)=1/8
所以是 1/4 + 1/8 + 1/16 + ... = ?

10.if m ( √3/2     - 1/2  )       find m^6
                1/2        √3/2

看不懂

11. find the coefficient od a^4bc in the expansion of (a+b+c+d)^6

因为 a^4bc 的排法有 6!/(4!x1!x1!) = 30 , 所以他的 coefficient = 30

12.if the lines 7x-5y+3 = 0 , 11x+2y+3=0,11x+2y+1=0 and 23x+23y+k=o are concurrent, find k.

concurrent = 交叉在同一个点。但是 11x+2y+3=0,11x+2y+1=0这两条线又 parellel ,很奇怪。题目有误？

13.if w is a complex cube root of unity , evaluate ( 1+ 3w+w^2)^2 + ( 1 + w+3w^2)^2

你必须知道 w^3 = 1 ,而且 w^2 + w + 1 = 0

14.if a = ∫1  1/ 1+x^3 dx , b = ∫1 1/1+x^4dx and c= ∫1 1/1+x^5dx.
                0                                0                                 0
     wat is the relationship between a, b & c ??

我不清楚你要什么 relationship , 我猜是 a<b<c ，因为

when 0=<x=<1 ==> x^3 >= x^4 >= x^5
所以 1/(1+x^3) =< 1/(1+x^4) =< 1/(1+x^5)
那么你在三个地方都拿 integral 的话就得到 a < b < c

jernie

还 有 一 题
given tat sin α+sinβ = 1/4 ,cosα  + cos β = 1/3 find the value of tan ( α+ β)


正 确 答 案 是  24/7  不 过 我 算 出 来 的 是  -5/12


应 该 怎 么 算

jernie

8.if 1-a tanθ=secθ  , b- cotθ  = cosec θ .find the value of a.b
shortform : sinθ = s , cosθ = c ,所以
1 - as/c = 1/c ==> c - as = 1 ...(1)
b - c/s = 1/s ==> bs - c = 1 ...(2)
(1) : a = (c-1)/s
(2) : b = (c+1)/s
=> a.b = (c^2-1)/s^2 = -s^2/s^2 = -1


QUOTE:
9. evaluate log9 √3 + log 9 √√3 + log 9 √√√3 + ....   (log 9 的 9是 底 数 )
注意到 log9 √3 = log9 9^(1/4) = 1/4,log 9 √√3=log9 9^(1/8)=1/8
所以是 1/4 + 1/8 + 1/16 + ... = ?


我 还 是 不 懂  

dunwan2tellu

Q8 你不懂哪里？我只是把 sin x ，cos x 写成 s,c 罢了。
然后用 sin^2 x+cos^2 x = 1 (也就是 s^2+c^2=1 ==> 1-c^2 = -s^2)

Q9
那么 square root 3 的。如果你 square root 1 次，就是 3^(1/2)
square root 2 次就是  3^(1/4)
3 次 = 3^(1/8)
以此类推，他的 power 是 1/2 ,1/4,1/8,1/16,1/32 ....

但是因为 base 是9，所以我又把 3 写成 9^(1/2)

example : log9 √√3 = log9 √√9^(1/2) = log9 9^(1/2 * 1/2 * 1/2 )
= log9 9^(1/8) = 1/8

ok 吗？

dunwan2tellu

given tat sin α+sinβ = 1/4 ,cosα  + cos β = 1/3 find the value of tan ( α+ β)

sinA+sinB=2sin(A+B/2)cos(A-B)/2 = 1/4
所以 sin(A+B/2)cos(A-B)/2 = 1/8 ....(1)
cosA+cosB=2cos(A+B/2)cos(A-B/2) = 1/3
所以 cos(A+B/2)cos(A-B/2) = 1/6 ....(2)
(1)/(2) ==> tan(A+B/2) = 3/4 ....(3)

由于
tan(A+B) = 2tan(A+B/2)/[1-(tan(A+B/2))^2] = 2(3/4)/(1-9/16) = 24/7

jernie

原帖由 jernie 于 14-8-2006 09:34 PM 发表
大 家 帮 帮 忙  教 教 我 怎 么 做

1. if the line y=3x+c is a tangent to the ellipse x^2+4y^2 = 4 ,find c.

2. find the parametric equation of the ellipse (x+1)^2 + 4 (y-1)^2 = 4

3. AB is a ...

question 3 how to do??