Maths problem: how yo do (b), thanks?

註：

其實另外那個帖（在英文版那個，標題為「Maths problem: would you mind to teach me how to do (b) i and ii, thanks?」）我也回答了，但我忘記了內容有「１０」的字樣而沒有由半形轉到全形字，所以被雅虎隱藏了。［之前 "Yahoo!管理員" 這位會員已經提醒了大家］所以只能回覆這個帖，幸好你發問了兩次，否則那個帖我的回答被隱藏了我的解答就無法再找地方告訴你。意見欄又取消了，雅虎現在不鼓勵會員之間多溝通，相信是避免其他問題。

English: Use gerund after "mind".

'Would you mind "teaching" me how to do?'

(a)

∠ACB = 90° (∠ in semi-circle)

tan∠CAB = CD/AD

tan∠CBD = CD/BD

tan(90°-∠CAD) = CD/BD

1/tan(∠CAD) = CD/BD

1/(CD/AD) = CD/BD

AD/CD = CD/BD

CD² = AD × BD

(b)

(i)

A = (4, 6), C = (x, y), so D = (4, y).

BD = x, so B = (4, y - x).

CD² = AD × BD

(x - 4)² = (6 - y) × x

x² - 8x + 16 = 6x - xy

x² - 14x + xy + 16 = 0

x² - (14 - y)x + 16 = 0 ... (*)

(ii)

From the diagram D(4, y) is vertically below A(4, 6), so y < 6. If y is an even number, then the largest possible value of y is 4.

For y = 4, (*) becomes

x² - 1０x + 16 = 0

(x - 2)(x - 8) = 0

x = 2 or x = 8

Note that C(x, y) is horizontally to the right of D(4, y), so x > 4.

Therefore, x = 8 and B = (4, y - x) = (4, -4) which is below the x-axis while A = (4, 6) is above the x-axis.

That is, the circle intersects the x-axis (at two points), so we do NOT agree with the student. The answer is NO.

The center of the circle is the mid-point of AB, i.e. (4, 1) and the radius of the circle is AB/2 = [6 - (-4)]/2 = 5.

The equation of the circle is therefore

(x - 4)² + (y - 1)² = 5²

x² + y² - 8x - 2y - 8 = 0.