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3.\;點與圓的關係

已知坐標平面上點A(5,4)與一圓
\C:x^2+y^2=4x，\;試求C上距離
\A點最近的點P的坐標。

A(5,4)

Q(2,0)

P

坐標平面上的圓
\C： \;(x-7)^2+(y-8)^2=9上有
\多少個點與原點的距離正好是整
\數值。

Q(7,8)

O

已知坐標平面上一圓
\C:(x-8)^2+(y-7)^2=9，\;有一
\光線由點A(4,2)出發，\;碰到y軸
\後反射到此圓C上的點Q，\;試求此
\光線行經的最短路線長。

A(4,2)

A'(-4,2)

Q(8,7)

R

P

P

d

A

切線段長

已知坐標平面上點P(2,3)為圓
\C:2x^2+2y^2+4x-4y+1=0
\外一點，\;自P點向圓作切線，\;試求
\切線段長。

P(2,3)

P(x_0,y_0)

A

B

Q

s = 0 (slider from 0 to max, step 1)
B = (15.06, -7.78)
poly_0, b, a, c, d = Polygon(B, B + (18, 0), B + (18, 12), B + (0, 12))
C = PointIn(poly_0)
O = DynamicCoordinates(C, floor(x(C)), floor(y(C)))
u = Vector((x(B), y(O)), (x(B) + 18, y(O)))
v = Vector((x(O), y(B)), (x(O), y(B) + 12))
max = 74
A: Corner[4] - (0, 1.6)
c_{12}: (x - 2 - x(O))^(2) + (y - y(O))^(2) = 4
A_{15}: O + (2, 0)
A_{12}: O + (5, 4)
text13: "" + (LaTeX[If[s ≥ 13, "x^2+y^2=4x"]]) + "" + (LaTeX[If[s ≥ 15, "⇒(x-2)^2+y^2=4"]]) + "
\\" + (LaTeX[If[s ≥ 16, "⇒圓心Q(2,0),r=2"]]) + "
\\" + (LaTeX[If[s ≥ 17, "∵\overline{AQ}=\sqrt{3^2+4^2}=5>2=r"]]) + "
\\" + (LaTeX[If[s ≥ 18, "∴A在圓外且\overline{PQ}=2,\overline{AP}=3"]]) + "
\\" + (LaTeX[If[s ≥ 19, "故由分點公式知"]]) + "
\\" + (LaTeX[If[s ≥ 20, "OP=\frac{2}{5}OA+\frac{3}{5}OQ"]]) + "
\\" + (LaTeX[If[s ≥ 21, "\;\;\;\;\;\;=\frac{2}{5}(5,4)+\frac{3}{5}(2,0)"]]) + "" + (LaTeX[If[s ≥ 22, "=(\frac{16}{5},\frac{8}{5})"]]) + "" + (LaTeX[If[s ≥ 23, "⇒P(\frac{16}{5},\frac{8}{5})"]]) + ""
e = Segment(A_{12}, A_{15})
D = Intersect(c_{12}, e, 1)
c_{24}: (x - 7 - x(O))^(2) + (y - y(O) - 8)^(2) = 9
A_{25}: O + (7, 8)
f = Segment(A_{25}, O)
text25: "圓C的圓心是Q(7,8),半徑r=3
\\" + (LaTeX[If[s ≥ 27, "\overline{OQ}=\sqrt{7^2+8^2}=\sqrt{113}≈10.6"]]) + "
\\" + (LaTeX[If[s ≥ 28, "最近距\overline{OQ}-r=\sqrt{113}-3≈7.6"]]) + "
\\" + (LaTeX[If[s ≥ 29, "最遠距\overline{OQ}+r=\sqrt{113}+3≈13.6"]]) + "
\\" + (LaTeX[If[s ≥ 30, "故7.6<d<13.6,"]]) + "" + (LaTeX[If[s ≥ 31, "又d為整數"]]) + "
\\" + (LaTeX[If[s ≥ 32, "∴d=8,9,10,11,12,13"]]) + "
\\" + (LaTeX[If[s ≥ 33, "以原點為圓心，\;d為半徑的圓與 \\圓C均交於兩點，\;"]]) + "" + (LaTeX[If[s ≥ 34, "故圓C上有12個 \\點符合與原點的距離為整數。"]]) + ""
g = Ray(O, A_{25})
E, F = Intersect(c_{24}, g)
h = Segment(F, O)
i = Segment(O, E)
list1 = Sequence(Circle[O, k + 7], k, 1, 6)
text5: "(2)\;已知平面上一點P(x_0,y_0)與圓
\\C:x^2+y^2+dx+ey+f=0,則
\\" + (LaTeX[If[s ≥ 6, "(a)若點P在圓的內部："]]) + "
\\" + (LaTeX[If[s ≥ 7, "\;\;\;\;\;x_0^2+y_0^2+dx_0+ey_0+f<0"]]) + "
\\" + (LaTeX[If[s ≥ 8, "(b)若點P在圓上："]]) + "
\\" + (LaTeX[If[s ≥ 9, "\;\;\;\;\;x_0^2+y_0^2+dx_0+ey_0+f＝0"]]) + "
\\" + (LaTeX[If[s ≥ 10, "(c)若點P在圓的外部："]]) + "
\\" + (LaTeX[If[s ≥ 11, "\;\;\;\;\;x_0^2+y_0^2+dx_0+ey_0+f>0"]]) + ""
c_{35} = Circle(O + (8, 7), 3)
A_{35}: O + (4, 2)
A_{36}: O + (8, 7)
A_{37}: O + ((-4), 2)
j = Line(A_{37}, A_{36})
A_{38} = Intersect(j, x = x(O))
G, H = Intersect(c_{35}, j)
k = Segment(A_{36}, A_{37})
text36: "∵圓C的圓心為Q(8,7),半徑r=3
\\" + (LaTeX[If[s ≥ 37, "由反射原理，\;取A關於y軸的對稱 \\點A'(-4,2)，\;"]]) + "" + (LaTeX[If[s ≥ 38, "連\overline{A'Q}交y軸於P點。 \\交圓C於R點。\;"]]) + ""
l = Segment(A_{38}, A_{35})
text39: "\;\;\;\;\overline{AP}+\overline{PR}" + (LaTeX[If[s ≥ 40, "\\=\overline{A'P}+\overline{PR}"]]) + "" + (LaTeX[If[s ≥ 41, "\\=\overline{AR}"]]) + "" + (LaTeX[If[s ≥ 42, "\\=\overline{A'Q}-\overline{QR}"]]) + "
\\" + (LaTeX[If[s ≥ 43, "=\sqrt{12^2+5^2}-3"]]) + "" + (LaTeX[If[s ≥ 44, "\\=13-3\\=10"]]) + ""
poly1, m = 工具1(I)
poly_a, n, p, q, r_1 = Polygon(Corner[1], Corner[2], Corner[3], Corner[4])
J = (25.24, 2.26)
r = 3 (slider from 1 to 5, step 0.1)
c_2 = Circle(J, r)
K = (17.76, 2.46)
t = Segment(K, J)
K_0 = Intersect(c_2, Ray[J, K])
a_1 = Segment(K_0, J)
A_3 = Mirror(K_0, J)
b_1 = Segment(J, A_3)
M_a: Midpoint[K, A_3] + (0, (0.2 * Segment[K, A_3]))
c_1 = CircumcircleArc(A_3, M_a, K)
M_b: Midpoint[K_0, K] + (0, (0.2 * Segment[K, K_0]))
d_1 = CircumcircleArc(K_0, M_b, K)
text1: "(1)\;點到圓的距離
\\" + (LaTeX[If[s ≥ 2, "設圓C的圓心為Q半徑為r，\;P為\\平面上任一點，\;如圖，\;A在圓C\\上且\overline{PQ}=d。"]]) + "
\\" + (LaTeX[If[s ≥ 3, "(a)若d>r，\;則\overline{AP}的最大值為 \\\;\;\;\;\;d+r，\;\;\;\;\;最小值為d-r；"]]) + "
\\" + (LaTeX[If[s ≥ 4, "(b)若d<r， \;則\overline{AP}的最大值為\\\;\;\;\;r+d，\;\;\;\;\;最小值為r-d；"]]) + ""
t_1: "Q" + (LaTeX[If[s ≥ 5, "(-\frac{d}{2},-\frac{e}{2})"]]) + ""
text14: "\overline{PQ}^2=(x_0+\frac{d}{2})^2+(y_0+\frac{e}{2})^2
\\=x_0^2+y_0^2+dx_0+ey_0+\frac{d^2}{4}+\frac{e^2}{4}\\" + (LaTeX[If[s ≟ 7, "<", If[s ≟ 9, "=", If[s ≟ 11, ">"]]]]) + "\frac{d^2+e^2-4f}{4}=r^2"
V_1: A + (0.4, (-9.8))
V_B: V_1 + (1.8, 0)
v_V = Segment(V_1, V_B)
poly_v, e_1, v_b, f_1 = Polygon(V_B, V_B - (0.3, 0), V_B + ((-0.3), 0.3))
L: A + (5, (-9.8))
g_1, poly2 = BigVector(L)
M: A + (9.8, (-9.8))
h_1, poly3 = BigVector(M)
text46: "坐標平面上，\;" + (LaTeX[If[s ≥ 47, "已知點P(x_0,y_0)"]]) + "" + (LaTeX[If[s ≥ 48, "為圓 \\C:x²+y²+dx+ey+f=0外 \\一點，\;"]]) + "" + (LaTeX[If[s ≥ 49, "自P對圓C作兩切線，\;"]]) + "" + (LaTeX[If[s ≥ 50, "其切 \\點為A,B，\;"]]) + "" + (LaTeX[If[s ≥ 51, "則切線段長\overline{PA}=\overline{PB} \\=\sqrt{x_0²+y_0^2+dx_0+ey_0+f}"]]) + ""
text52: "" + (LaTeX[If[(s ≥ 52) ∧ (s ≤ 57), "證明"]]) + "
\\" + (LaTeX[If[(s ≥ 53) ∧ (s ≤ 57), "圓的圓心為Q(-\frac{d}{2},=\frac{e}{2}),半徑為 \\r=\frac{1}{2}\sqrt{d^2+e^2-4f}"]]) + "
\\" + (LaTeX[If[(s ≥ 54) ∧ (s ≤ 57), "因為切線與半徑垂直"]]) + "
\\" + (LaTeX[If[(s ≥ 55) ∧ (s ≤ 57), "所以\Delta PAQ 為直角三角形"]]) + "
\\" + (LaTeX[If[s ≥ 56, "又\overline{PQ}^2=(x_0+\frac{d}{2})^2+(y_0+\frac{e}{2})^2"]]) + "
\\" + (LaTeX[If[s ≥ 57, "\overline{PA}^2=\overline{PQ}^2-r^2"]]) + "
\\" + (LaTeX[If[s ≥ 59, "=(x_0+\frac{d}{2})^2+(y_0+\frac{e}{2})^2 \\\;\;\;\;\;\;\;\;\;\;-\frac{d^2+e^2-4f}{4}"]]) + "
\\" + (LaTeX[If[s ≥ 60, "=x_0^2+y_0^2+dx_0+ey_0+f"]]) + "
\\" + (LaTeX[If[s ≟ 61, "\overline{PA}=\sqrt{x_0^2+y_0^2+dx_0+ey_0+f}"]]) + ""
c_{62}: (x - x(O))^(2) + (y - y(O))^(2) + (2 * (x - x(O))) - (2 * (y - y(O))) + 0.5 = 0
P_{62}: O + (2, 3)
i_1, j_1 = Tangent(P_{62}, c_{62})
N = Intersect(c_{62}, i_1)
P = Intersect(c_{62}, j_1)
Q = (25.46, 2.02)
R = (21.12, -1.48)
c_{48} = Circle(R, 3)
k_1, l_1 = Tangent(Q, c_{48})
T = Intersect(c_{48}, k_1)
U = Intersect(c_{48}, l_1)
m_1 = Segment(T, Q)
n_1 = Segment(Q, U)
p_1 = Segment(T, R)
q_1 = Segment(R, U)
s_1 = Segment(Q, R)
text63: "" + (LaTeX[If[(s ≥ 63) ∧ (s ≤ 69), "法一"]]) + "
\\" + (LaTeX[If[(s ≥ 64) ∧ (s ≤ 69), "C:\;x²+y²+2x-2y+\frac{1}{2}=0"]]) + "
\\" + (LaTeX[If[(s ≥ 65) ∧ (s ≤ 69), "⇒(x+1)²+(y-1)²=\frac{3}{2}"]]) + "
\\" + (LaTeX[If[(s ≥ 66) ∧ (s ≤ 69), "⇒Q(-1,1),r^2=\frac{3}{2}"]]) + "
\\" + (LaTeX[If[(s ≥ 67) ∧ (s ≤ 69), "又\overline{PQ}^2=3^2+2^2=13"]]) + "
\\" + (LaTeX[If[(s ≥ 68) ∧ (s ≤ 69), "故切線段長為\sqrt{13-\frac{3}{2}}"]]) + "" + (LaTeX[If[s ≟ 69, "=\frac{\sqrt{46}}{2}"]]) + "" + (LaTeX[If[s ≥ 70, "法二"]]) + "
\\" + (LaTeX[If[s ≥ 71, "C:\;x²+y²+2x-2y+\frac{1}{2}=0"]]) + "
\\" + (LaTeX[If[s ≥ 72, "將P點代入切線段長公式得"]]) + "
\\" + (LaTeX[If[s ≥ 73, "\sqrt{2²+3²+2×2-2×3+\frac{1}{2}}"]]) + "
\\" + (LaTeX[If[s ≥ 74, "=\frac{\sqrt{46}}{2}"]]) + ""
Q_{66}: O + (-1, 1)
a_2 = Segment(N, Q_{66})
b_2 = Segment(Q_{66}, P)
d_2 = Segment(P_{62}, Q_{66})
α = Angle(Q, T, R)
β = Angle(R, U, Q)
e_2 = Segment(P_{62}, N)
f_2 = Segment(P_{62}, P)

Tool: 工具1
工具1[ <點> ]
Input: V_1
Output: Poly_V, v_1

V_2: V_1 + (0.8, 0)
V_3: V_1 + (0.6, 0.2)
V_4: V_1 + (0.6, 0)
Poly_V, v_4, v_2, v_1 = Polygon(V_2, V_3, V_4)

Tool: Big Vector (command BigVector)
BigVector[ <點> ]
Input: V_1
Output: v_V, poly_v

V_B: V_1 + (1.8, 0)
v_V = Segment(V_1, V_B)
poly_v, e_1, v_b, f_1 = Polygon(V_B, V_B - (0.3, 0), V_B + ((-0.3), 0.3))