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4.\;直線與圓的關係

已知平面上直線L:x-y+k=0
\與圓C:x^2+y^2=2，\;試就下列關
\係討論k的範圍：
\(1)\;直線L與圓C交於兩點；
\(2)\;直線L與圓C交於一點；
\(3)\;直線L與圓C沒有交點；

$k=%v$

設圓C:(x-2)^2+(y+3)^2=10，
\試求通過點P(-3,2)且與C相切之
\直線方程式。

P(-3,2)

Q(2,-3)

(x-2)^2+(y-3)^2=10

設圓C:(x-1)^2+(y-3)^2=9，
\試求通過點P(-2,1)且與C相切之
\直線方程式。

P(-2,1)

Q(1,3)

求與圓C:(x+5)^2+(y-1)^2=20
\相切且斜率為2之直線方程式。

$k=%v$

過圓上一點作圓的切線

已知坐標平面上一點P(x_0,y_0)在
\圓C:\;x^2+y^2+dx+ey+f=0
\上，則過P之切線方程式為
\L:\;x_0x+y_0y+d·\frac{x+x_0}{2}
\\;\;\;\;\;\;\;\;+e·\frac{y+y_0}{2}+f=0

P(x_0,y_0)

Q

已知坐標平面上一點P(5,-6)，\;且
\P在圓x^2+y^2-4x+6y-5=0
\上，\;試求通過點P的切線方程式。

P(5,-6)

Q(2,-3)

(4,2)

(5,1)

(-2,2)

(-2,2)

設\sqrt{x(4-x)}=mx+4有相異兩
\實數解，\;試求m之範圍。

$m=%v$

Q(2,0)

(0,4)

r = 2.1999999999999997
Q_{99} = (27.22, 1.06)
s = 0 (slider from 0 to max, step 1)
B = (15.66, -6.54)
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 = 157
poly_a, e, f, g, h = Polygon(Corner[1], Corner[2], Corner[3], Corner[4])
A: Corner[4] - (0, 1)
text1: "" + (LaTeX[If[(s ≥ 1) ∧ (s ≤ 4), "(1)\;已知圓C:(x-h)^2+(y-k)^2=r^2，\;直線L:ax+by+c=0，\; \\\;\;\;\;\;\;若圓心為Q(h,k)，\;且以d(Q,L)表Q點到直線L的距離，\;則："]]) + "" + (LaTeX[If[(s ≥ 2) ∧ (s ≤ 4), "\\\;\;\;\;\;\;(a)\;d(Q,L)<r時，\;直線L與圓C交於兩點(相割)"]]) + "" + (LaTeX[If[s ≟ 3 ∨ s ≟ 4, "\\\;\;\;\;\;\;(b)\;d(Q,L)=r時，\;直線L與圓C交於一點(相切)"]]) + "" + (LaTeX[If[s ≟ 4, "\\\;\;\;\;\;\;(c)\;d(Q,L)>r時，\;直線L與圓C沒有交點(相離)"]]) + "" + (LaTeX[If[s ≥ 5, "(2)\;已知圓C:x^2+y^2+dx+ey+f=0，\;直線L:ax+by+c=0，\;"]]) + "
\\\;\;\;\;\;\;" + (LaTeX[If[s ≥ 6, "將y=-\frac{ax+c}{b}代入圓C中，\;"]]) + "" + (LaTeX[If[s ≥ 7, "得到一個x的二次式： \\\;\;\;\;\;\;Ax^2+Bx+c=0，\;"]]) + "" + (LaTeX[If[s ≥ 8, "令上式的判別式為D=B^2-4AC，\;則："]]) + "
\\\;\;\;\;\;\;" + (LaTeX[If[s ≥ 9, "(a)\;D>0時，\;直線L與圓C交於兩點(相割)"]]) + "
\\\;\;\;\;\;\;" + (LaTeX[If[s ≥ 10, "(b)\;D=0時，\;直線L與圓C交於一點(相切)"]]) + "
\\\;\;\;\;\;\;" + (LaTeX[If[s ≥ 11, "(c)\;D<0時，\;直線L與圓C沒有交點(相離)"]]) + ""
list1: {"x-y+k=0", "x^2+y^2=2"}
c_{12} = Circle(O, 1.4142135623730951)
k = 4 (slider from -4 to 4, step 0.001)
L_{12}: x - x(O) - (y - y(O)) + k = 0
text3: "x-y" + (LaTeX[If[k  <  0, "-" + (abs(k)), ("+" * k)]]) + "=0"
text13: "" + (LaTeX[If[s ≤ 19, "法一"]]) + "
\\" + (LaTeX[If[(s ≥ 14) ∧ (s ≤ 19), ("考慮" * TableText[{list1}, "v{"])]]) + "
\\" + (LaTeX[If[(s ≥ 15) ∧ (s ≤ 19), "將y=x+k代入C中，\;得："]]) + "
\\" + (LaTeX[If[(s ≥ 16) ∧ (s ≤ 19), "x^2+(x+k)^2=2"]]) + "" + (LaTeX[If[(s ≥ 17) ∧ (s ≤ 19), "\\⇒2x^2+2kx+(k^2-2)=0"]]) + "
\\" + (LaTeX[If[(s ≥ 18) ∧ (s ≤ 19), "D=(2k)^2-4×2×(k^2-2)"]]) + "
\\" + (LaTeX[If[s ≥ 19, "⇒D=-4(k^2-4)"]]) + "" + (LaTeX[If[s ≥ 21, "\\\;\;\;\;\;\;\;\;\;=-4(k+2)(k-2)"]]) + "
\\" + (LaTeX[If[(s ≥ 22) ∧ (s ≤ 26), "(1)\;相割：\;"]]) + "" + (LaTeX[If[(s ≥ 23) ∧ (s ≤ 26), "D>0"]]) + "
\\" + (LaTeX[If[(s ≥ 24) ∧ (s ≤ 26), "⇒-4(k+2)(k-2)>0"]]) + "
\\" + (LaTeX[If[(s ≥ 25) ∧ (s ≤ 26), "⇒(k+2)(k-2)<0"]]) + "
\\" + (LaTeX[If[s ≟ 26, "⇒-2<k<2"]]) + "
\\" + (LaTeX[If[(s ≥ 27) ∧ (s ≤ 30), "(2)\;相切：\;"]]) + "" + (LaTeX[If[(s ≥ 28) ∧ (s ≤ 30), "D=0"]]) + "
\\" + (LaTeX[If[(s ≥ 29) ∧ (s ≤ 30), "⇒-4(k+2)(k-2)=0"]]) + "
\\" + (LaTeX[If[s ≟ 30, "⇒k=2,-2"]]) + "
\\" + (LaTeX[If[s ≥ 31, "(3)\;相離：\;"]]) + "" + (LaTeX[If[s ≥ 32, "D<0"]]) + "
\\" + (LaTeX[If[s ≥ 33, "⇒-4(k+2)(k-2)<0"]]) + "
\\" + (LaTeX[If[s ≥ 34, "⇒(k+2)(k-2)>0"]]) + "
\\" + (LaTeX[If[s ≥ 35, "⇒k<-2或k>2"]]) + ""
text36: "法二
\\" + (LaTeX[If[s ≥ 37, "圓C的圓心為O(0,0)半徑為\sqrt{2}，\;"]]) + "" + (LaTeX[If[s ≥ 38, " \\考慮O到直線L的距離"]]) + "
\\" + (LaTeX[If[s ≥ 39, "d(O,L)=\frac{|0-0+k|}{\sqrt{1^2+1^2}}"]]) + "" + (LaTeX[If[s ≥ 40, "=\frac{|k|}{\sqrt{2}}"]]) + "
\\" + (LaTeX[If[(s ≥ 41) ∧ (s ≤ 45), "(1)\;相割：\;"]]) + "" + (LaTeX[If[(s ≥ 42) ∧ (s ≤ 45), "d(O,L)<r"]]) + "
\\" + (LaTeX[If[(s ≥ 43) ∧ (s ≤ 45), "⇒\frac{|k|}{\sqrt{2}}<\sqrt{2}"]]) + "" + (LaTeX[If[s ≟ 44 ∨ s ≟ 45, "⇒|k|<2"]]) + "
\\" + (LaTeX[If[s ≟ 45, "⇒-2<k<2"]]) + "
\\" + (LaTeX[If[(s ≥ 46) ∧ (s ≤ 50), "(2)\;相切：\;"]]) + "" + (LaTeX[If[(s ≥ 47) ∧ (s ≤ 50), "d(O,L)=r"]]) + "
\\" + (LaTeX[If[(s ≥ 48) ∧ (s ≤ 50), "⇒\frac{|k|}{\sqrt{2}}=\sqrt{2}"]]) + "" + (LaTeX[If[(s ≥ 49) ∧ (s ≤ 50), "⇒|k|=2"]]) + "
\\" + (LaTeX[If[s ≟ 50, "⇒k=2或-2"]]) + "
\\" + (LaTeX[If[s ≥ 51, "(3)\;相離：\;"]]) + "" + (LaTeX[If[s ≥ 52, "d(O,L)>r"]]) + "
\\" + (LaTeX[If[s ≥ 53, "⇒\frac{|k|}{\sqrt{2}}>\sqrt{2}"]]) + "" + (LaTeX[If[s ≥ 54, "⇒|k|>2"]]) + "
\\" + (LaTeX[If[s ≟ 55, "⇒k<-2或k>2"]]) + ""
D, E = Intersect(c_{12}, L_{12})
A_{56}: O + ((-3), 2)
c_{56} = Circle(O + (2, (-3)), 3.1622776601683795)
i, j = Tangent(A_{56}, c_{56})
Q_{57}: O + (2, (-3))
text57: "" + (LaTeX[If[(s ≥ 57) ∧ (s ≤ 64), "∵圓心Q(2,-3),半徑r=\sqrt{10}，\;"]]) + "
\\" + (LaTeX[If[(s ≥ 58) ∧ (s ≤ 64), "∴\overline{PQ}=\sqrt{5^2+5^2}=5\sqrt{2}>5=r"]]) + "
\\" + (LaTeX[If[(s ≥ 59) ∧ (s ≤ 64), "所以P在圓外，\;"]]) + "" + (LaTeX[If[(s ≥ 60) ∧ (s ≤ 64), "故令切線方程式為 \\L:y-2=m(x+3)"]]) + "
\\" + (LaTeX[If[(s ≥ 61) ∧ (s ≤ 64), "⇒y=mx+(3m+2)代入C得:"]]) + "
\\" + (LaTeX[If[(s ≥ 62) ∧ (s ≤ 64), "(x-2)^2+(mx+3m+5)^2=10 "]]) + "
\\" + (LaTeX[If[(s ≥ 63) ∧ (s ≤ 65), "(m^2+1)x^2+(6m^2+10m-4)x \\\;\;\;\;\;\;\;\;\;\;\;+(9m^2+30m+19)=0"]]) + "
\\" + (LaTeX[If[(s ≥ 64) ∧ (s  <  71), "因為圓C與直線L相切，\;"]]) + "" + (LaTeX[If[(s ≥ 66) ∧ (s  <  71), "所以上式 \\之判別式D=0"]]) + "
\\" + (LaTeX[If[(s ≥ 67) ∧ (s  <  71), "D=(6m^2+10m-4)^2 \\\;\;\;\;\;-4(m^2+1)(9m^2+30m+19)"]]) + "
\\" + (LaTeX[If[(s ≥ 68) ∧ (s  <  71), "整理得：\;3m^2+10m+3=0"]]) + "
\\" + (LaTeX[If[(s ≥ 69) ∧ (s  <  71), "⇒(3m+1)(m+3)=0"]]) + "
\\" + (LaTeX[If[s ≥ 70, "⇒m=-3,-\frac{1}{3}"]]) + "
\\" + (LaTeX[If[s ≥ 72, "故所求之切線方程式為: \\y-2=3(x+3) ,\\y-2=-\frac{1}{3}(x+3)"]]) + "
\\" + (LaTeX[If[s ≥ 73, "整理得： \\3x+y+7=0, \\x+3y-3=0"]]) + ""
c_{74} = Circle(O + (1, 3), 3)
P_{74}: O + ((-2), 1)
Q_{75}: O + (1, 3)
l, m = Tangent(P_{74}, c_{74})
text75: "" + (LaTeX[If[(s ≥ 75) ∧ (s ≤ 81), "∵圓C的圓心為Q(1,3),半徑r=3"]]) + "
\\" + (LaTeX[If[(s ≥ 76) ∧ (s ≤ 81), "又\overline{PQ}=\sqrt{3^2+1^2}=\sqrt{10}>3=r"]]) + "
\\" + (LaTeX[If[(s ≥ 77) ∧ (s ≤ 81), "過P與圓C相切的直線應有兩條"]]) + "
\\" + (LaTeX[If[(s ≥ 78) ∧ (s ≤ 81), "設切線方程式為 \\L:y-1=m(x+2)"]]) + "
\\" + (LaTeX[If[(s ≥ 79) ∧ (s ≤ 81), "⇒mx-y+(2m+1)=0"]]) + "
\\" + (LaTeX[If[(s ≥ 80) ∧ (s ≤ 81), "考慮圓心到切線的距離等於半徑"]]) + "
\\" + (LaTeX[If[s ≥ 81, "d(Q,L)=\frac{|m-3+2m+1|}{\sqrt{m^2+1}}=3"]]) + "
\\" + (LaTeX[If[s ≥ 83, "⇒|3m-2|=3\sqrt{m^2+1}"]]) + "
\\" + (LaTeX[If[s ≥ 84, "9m^2-12m+4=9m^2+9"]]) + "
\\" + (LaTeX[If[s ≥ 85, "⇒12m=-5⇒m=-\frac{5}{12}"]]) + "
\\" + (LaTeX[If[s ≥ 86, "此處m只有一解，\;故另一條切線\\斜率不存在(鉛直)"]]) + "" + (LaTeX[If[s ≥ 87, ",故切線方程式:\\y-1=\frac{5}{12}(x+2)與x=-1"]]) + "
\\" + (LaTeX[If[s ≥ 88, "5x+12y-2=0與x=-1"]]) + ""
c_{89} = Circle(O + ((-5), 1), 4.47213595499958)
text90: "\\圓心Q(-5,1),半徑r=2\sqrt{5}
\\" + (LaTeX[If[s ≥ 91, "設切線方程式為L：\;y=2x+k"]]) + "
\\" + (LaTeX[If[s ≥ 92, "⇒2x-y+k=0\;"]]) + "" + (LaTeX[If[s ≥ 93, "又d(Q,L)=r"]]) + "
\\" + (LaTeX[If[s ≥ 94, "故\frac{|2×(-5)-1+k|}{\sqrt{2^2+(-1)^2}}=2\sqrt{5}"]]) + "
\\" + (LaTeX[If[s ≥ 95, "⇒|k-11|=10"]]) + "" + (LaTeX[If[s ≥ 96, "⇒k=1或21"]]) + "
\\" + (LaTeX[If[s ≥ 97, "L:2x-y+1=0,2x-y+21=0"]]) + ""
n = 0.9 (slider from -20 to 30, step 0.1)
L_{91}: (2 * (x - x(O))) - (y - y(O)) + n = 0
L_{97}: (2 * (x - x(O))) - (y - y(O)) + 1 = 0
L_{98}: (2 * (x - x(O))) - (y - y(O)) + 21 = 0
c_{99} = Circle(Q_{99}, r)
P_{99} = Point(c_{99})
t_{99} = Segment(P_{99}, Q_{99})
L_{99} = OrthogonalLine(P_{99}, t_{99})
c_{113}: (x - x(O))^(2) + (y - y(O))^(2) - (4 * (x - x(O))) + (6 * (y - y(O))) - 5 = 0
P_{113}: O + (5, (-6))
L_{113} = Tangent(P_{113}, c_{113})
F: A + (1.7, (-7.9))
p, poly1 = BigVector(F)
Q_{113}: O + (2, (-3))
text121: "法二
\\" + (LaTeX[If[s ≥ 122, "L:5x+(-6)y-4·\frac{x+5}{2} \\\;\;\;\;\;+6·\frac{y+(-6)}{2}-5=0"]]) + "
\\" + (LaTeX[If[s ≥ 123, "⇒3x-3y+33=0"]]) + "
\\" + (LaTeX[If[s ≥ 124, "⇒x-y+11=0"]]) + ""
w = Vector(P_{113}, Q_{113})
text114: "法一
\\" + (LaTeX[If[s ≥ 115, "圓心Q(2,-3),半徑r=3\sqrt{2}"]]) + "
\\" + (LaTeX[If[s ≥ 116, "取PQ=(-3,3)=-3(1,-1)為 \\切線的法向量"]]) + "
\\" + (LaTeX[If[s ≥ 117, "設切線為L:x-y=k"]]) + "
\\" + (LaTeX[If[s ≥ 118, "又P在其上"]]) + "" + (LaTeX[If[s ≥ 119, "⇒k=5-(-6)=11"]]) + "
\\" + (LaTeX[If[s ≥ 120, "⇒L:x-y=11"]]) + ""
G: A + (1.6, (-8.1))
q, poly2 = BigVector(G)
text100: "" + (LaTeX[If[(s ≥ 99) ∧ (s ≤ 102), "證明"]]) + "
\\" + (LaTeX[If[(s ≥ 100) ∧ (s ≤ 102), "圓心Q(-\frac{d}{2},-\frac{e}{2}),\\半徑r=\frac{1}{2}\sqrt{d^2+e^2-4f}"]]) + "
\\" + (LaTeX[If[(s ≥ 101) ∧ (s ≤ 102), "取QP=(x_0+\frac{d}{2},y_0+\frac{e}{2})為切線\\L的法向量"]]) + "
\\" + (LaTeX[If[(s ≥ 102) ∧ (s ≤ 109), "⇒L:(x_0+\frac{d}{2})x+(y_0+\frac{e}{2})y=k"]]) + "
\\" + (LaTeX[If[(s ≥ 104) ∧ (s ≤ 109), "因為P在L上"]]) + "" + (LaTeX[If[(s ≥ 105) ∧ (s ≤ 109), "∴k=x_0^2+\frac{d}{2}x_0+y_0^2+\frac{e}{2}y_0"]]) + "
\\" + (LaTeX[If[s ≥ 106, "⇒L:(x_0+\frac{d}{2})x+(y_0+\frac{e}{2})y=x_0^2+\frac{d}{2}x_0+y_0^2+\frac{e}{2}y_0"]]) + "
\\" + (LaTeX[If[(s ≥ 107) ∧ (s ≤ 109), "⇒x_0x+y_0y+\frac{d}{2}(x+x_0)+\frac{e}{2}(y+y_0)=x_0^2+dx_0+y_0^2+ey_0"]]) + "
\\" + (LaTeX[If[(s ≥ 108) ∧ (s ≤ 109), "又P在圓C上:x_0^2+y_0^2+dx_0+ey_0+f=0"]]) + "
\\" + (LaTeX[If[s ≥ 109, "⇒x_0^2+y_0^2+dx_0+ey_0=-f"]]) + "
\\" + (LaTeX[If[s ≥ 111, "⇒x_0x+y_0y+\frac{d}{2}(x+x_0)+\frac{e}{2}(y+y_0)=-f"]]) + "" + (LaTeX[If[s ≥ 112, "\\⇒L:x_0x+y_0y+\frac{d}{2}(x+x_0)+\frac{e}{2}(y+y_0)+f=0"]]) + ""
list2: {"x^2+y^2-2x+4y-20=0", "y=6-x"}
text125: "" + (LaTeX[If[s ≟ 125, "設"]]) + "" + (LaTeX[If[s ≥ 125, "C:x^2+y^2-2x+4y-20=0"]]) + ",
\\" + (LaTeX[If[s ≟ 125, "試判斷圓C與下列直線的關係，\;當 \\相割時，\;求交點坐標；\;當相切時， \\求切點坐標。"]]) + "
\\" + (LaTeX[If[(s ≥ 125) ∧ (s ≤ 133), "(1)\;L_1:x+y=6;"]]) + "
\\" + (LaTeX[If[s ≟ 125 ∨ (s ≥ 134), "(2)\;L_2:3x-4y+14=0"]]) + ""
c_{126} = Circle(O + (1, (-2)), 5)
L_{133} = Line(O + (4, 2), O + (5, 1))
P_{133}: O + (4, 2)
Q_{133}: O + (5, 1)
list3: {"x^2+y^2-2x+4y-20=0", "y=\frac{3}{4}x+\frac{7}{2}"}
text126: "" + (LaTeX[If[(s ≥ 127) ∧ (s ≤ 133), "(1)" + (TableText[{list2}, "v{"])]]) + "
\\" + (LaTeX[If[(s ≥ 128) ∧ (s ≤ 133), "x^2+(6-x)^2-2x+4(6-x)=20"]]) + "
\\" + (LaTeX[If[(s ≥ 129) ∧ (s ≤ 133), "⇒x^2-9x+20=0"]]) + "
\\" + (LaTeX[If[(s ≥ 130) ∧ (s ≤ 133), "⇒(x-4)(x-5)=0"]]) + "
\\" + (LaTeX[If[(s ≥ 131) ∧ (s ≤ 133), "⇒x=4,5"]]) + "" + (LaTeX[If[(s ≥ 132) ∧ (s ≤ 133), "⇒y=2,1"]]) + "
\\" + (LaTeX[If[s ≟ 133, "故L_1與C相割且交點為(4,2),(5,1)"]]) + "" + (LaTeX[If[s ≥ 135, "(2)" + (TableText[{list3}, "v{"])]]) + "
\\" + (LaTeX[If[(s ≥ 137) ∧ (s ≤ 140), "x^2+(\frac{3}{4}x+\frac{7}{2})^2-2x\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+4(\frac{3}{4}x+\frac{7}{2})=20"]]) + "
\\" + (LaTeX[If[(s ≥ 138) ∧ (s ≤ 140), "⇒\frac{25}{16}x^2+\frac{25}{4}x+\frac{25}{4}=0"]]) + "
\\" + (LaTeX[If[(s ≥ 139) ∧ (s ≤ 140), "⇒x^2+4x+4=0"]]) + "
\\" + (LaTeX[If[s ≥ 140, "⇒(x+2)^2=0"]]) + "
\\" + (LaTeX[If[s ≥ 142, "⇒x=-2"]]) + "" + (LaTeX[If[s ≥ 143, "⇒y=-2"]]) + "
\\" + (LaTeX[If[s ≥ 144, "故C與L_2相切於點(-2,2)"]]) + ""
P_{144}: O + ((-2), 2)
L_{144}: (3 * (x - x(O))) - (4 * (y - y(O))) + 14 = 0
list4: {"y=\sqrt{x(4-x)}", "y=mx+4"}
m_{151} = -0.75 (slider from -3 to 0, step 0.05)
f_{149}(x) = sqrt((4 * (x - x(O))) - (x - x(O))^(2)) + y(O)
f_{151}(x) = (m_{151} * (x - x(O))) + 4 + y(O)
Q_{150}: O + (2, 0)
P_{151}: O + (0, 4)
text146: "" + (LaTeX[If[(s ≥ 146) ∧ (s ≤ 153), TableText[{list4}, "v{"]]]) + "
\\" + (LaTeX[If[(s ≥ 147) ∧ (s ≤ 150), "其中y=\sqrt{4x-x^2}"]]) + "
\\" + (LaTeX[If[(s ≥ 148) ∧ (s ≤ 150), "⇒x^2+y^2-4x=0"]]) + "
\\" + (LaTeX[If[(s ≥ 149) ∧ (s ≤ 150), "⇒(x-2)^2+y^2=4,y≥0"]]) + "
\\" + (LaTeX[If[s ≟ 150, "表以(2,0)為圓心且 半徑為2之上半圓"]]) + "
\\" + (LaTeX[If[(s ≥ 151) ∧ (s ≤ 153), "y=mx+4表斜率為m且過(0,4)\\之直線L"]]) + "" + (LaTeX[If[(s ≥ 152) ∧ (s ≤ 153), "\;\;故我們考慮過(0,4)上半 \\圓的切線L_1"]]) + "
\\" + (LaTeX[If[s ≥ 153, "∵d(Q,L)=\frac{|2m+4|}{\sqrt{m^2+1^2}}=2=r"]]) + "
\\" + (LaTeX[If[s ≥ 154, "∴(m+2)^2=m^2+1"]]) + "" + (LaTeX[If[s ≥ 155, "⇒m=-\frac{3}{4}"]]) + "
\\" + (LaTeX[If[s ≥ 156, "而當直線通過(4,0)與 (0,4)時，\;直\\線與上半圓交於兩點 "]]) + "" + (LaTeX[If[s ≥ 157, "\;\;故當\\-1≤m<-\frac{3}{4}時，\\\sqrt{x(4-x)}=mx+4有相異兩實根"]]) + ""

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))