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第 17 週 · 坐標幾何 · 第 3 節

圓方程(標準式 · 一般式 · 配方還原)

直線講完,輪到圓。圓就是「到圓心距離都等於 r 的點」,用距離公式一寫就得到圓方程。這節把標準式 (x−a)²+(y−b)²=r²、一般式 x²+y²+Dx+Ey+F=0,和「配方法還原」講透,一眼讀出圓心半徑。配滿坐標圖例。

▶ 第一關: 標準式與圓心半徑
(a, b)(x, y)rdistance to center = r
(a, b)r(x-a)2 + (y-b)2 = r2center (a,b), radius r
(2, 3)r = 5(x-2)2 + (y-3)2 = 25center (2,3), r = sqrt(25) = 5
(x + 1)2=(x - (-1))2x=-1plus sign inside → center coord is negative(y + 4)2 → center y = -4
▶ 第二關: 符號陷阱與寫方程
(-3, 5)(x+3)2+(y-5)2=49(4, -2)(x-4)2+(y+2)2=16
(2, -3)r = 4(x-2)2 + (y+3)2 = 16center (2,-3), r=4, right side = 4^2 = 16
▶ 第三關: 一般式與配方還原
(x-2)2 + (y-3)2 = 4expand →x2 + y2 - 4x - 6y + 9 = 0standard form → general form
x2 + y2 - 4x - 6y + 9 = 0group: (x2-4x) + (y2-6y) + 9 = 0(x-2)2 - 4 + (y-3)2 - 9 + 9 = 0(x-2)2 + (y-3)2 = 4(2,3)r=2
x2 + y2 + Dx + Ey + F = 0center = (-D/2, -E/2)r from completing squareD=-4, E=-6 → center (2, 3)
⚔ BOSS: 一般式配方求圓心半徑
x2 + y2 - 10x + 9 = 0 → (x-5)2 + y2 = 16(5, 0)r = 4complete square in x → center (5,0), r = sqrt(16) = 4