F.5 Maths:More About Equations

用戶91174

2010-11-28 6:05 pm

A straight line intersects the x-axis and the y-axis at A(a,0) and B(0,b)respectively, and passes through (2,3). If the area of triangle △OAB(O is the origin) is 12.25, find all possible equations of the line AB.

Answer: 2x+y-7=0, 9x+8y-42=0

Please show the steps. Thank you.

回答 (1)

蛙人

2010-11-28 6:57 pm

✔ 最佳答案

Slope of AB = -b/a = (3-0)/(2-a)
ab-2b=3a----------------(1)
Area of triangle = 12.25
(1/2)(b)(a) = 12.25
2ab = 49------------------(2)
Hence by(1),
(49/2) - 2b = 3a
49 - 4b = 6a
a = (49 - 4b)/6
Sub a in (2),
2[(49-4b)/6]b = 49
4b^2 - 49b + 147 = 0
b = 7 or 21/4

Slope of AB = (7 - 3)/(0 - 2) or (21/4 - 3)/(0 - 2)
= -2 or -9/8
Equation of AB is
y - 3 = -2(x - 2) or y - 3 = (-9/8)(x - 2)
2x + y - 7 = 0 or 9x + 8y - 42 = 0

2010-11-28 11:01:04 補充：
留意只有當b及a皆是正數時,這兩條方程才是答案。當其中一個為負數的時候,每種可能會有多1條方程。如果b及a設定為任意實數的話,這道題目會有總共4個答案。

參考: me

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