數學問題 4 條????

1. (a)
P(x,y) divides the line segment in the ratio 2 : 5.
Thus, r = 2/5.

x = [x1 + rx1] / [1 + r]
x = [0 + (2/5)(-21)] / [1 + (2/5)]
x = -6

y = [y1 + ry2] / [1 + r]
y = [7 + (2/5)(0)] / [1 + (2/5)]
y = 5

Ans: The point is (-6, 5).


1. (b)
P(x,y) divides the line segment in the ratio 1 : 4/3.
Thus r = 3/4.

x = [x1 + rx1] / [1 + r]
x = [0 + (3/4)(-21)] / [1 + (3/4)]
x = -9

y = [y1 + ry2] / [1 + r]
y = [7 + (3/4)(0)] / [1 + (3/4)]
y = 4

Ans: The point is (-9, 4).

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2.
AP : PB = 2 : 1. Thus, r = 2.

3 = [x + 2(6)] / [1 + 2]
Hence, x = -3

2 = [(-1) + 2y] / [1 + 2]
Hence, y = 7/2

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3.
Let the coordinates of C be (x3, y3)

AB : BC = 2 : 3.
Thus AC : CB = 5 : -3, and r = -5/3

x3 = [0 + (-5/3)p] / [1 + (-5/3)]
x3 = 5p/2

y3 = [k + (-5/3)q] / [1 + (-5/3)]
y3 = (5q - 3k)/2

Hence, C is (5p/2, (5q - 3k)/2).


Let the coordinates of D be (x4, y4)

AB : BD = 2 : 7.
Thus AD : DB = 9 : -7, and r = -9/7

x4 = [0 + (-9/7)p] / [1 + (-9/7)]
x3 = 9p/2

y4 = [k + (-9/7)q] / [1 + (-9/7)]
y3 = (9q - 7k)/2

Hence, D is (9p/2, (9q - 7k)/2).

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4.
The mth point of division (x, y) divides the line segment in ratio m : (n-m).
Thus, r = m/(n - m)

x = [x1 + rx1] / [1 + r]
x = [a + (m/(n - m))c] / [1 + (m/(n - m))]
x = (an - am + cm)/n

y = [y1 + ry2] / [1 + r]
x = [b + (m/(n - m))d] / [1 + (m/(n - m))]
x = (bn - bm + dm)/n

Ans: The required point ((an - am + cm)/n, (bn - bm + dm)/n).