두 원의 공통 접선과 현의 길이 조건 문제

문제

좌표평면 위에 두 원 $C_1: x^2+y^2=1$과 $C_2: (x-3)^2+y^2=4$가 있다. 점 $P(a,b)$에서 그은 직선 $L$이 원 $C_1$에 접하고, 원 $C_2$와 만나는 두 점을 $A, B$라 할 때, 선분 $AB$의 길이가 $2\sqrt{3}$이다. 이러한 조건을 만족하는 서로 다른 직선 $L$이 정확히 두 개 존재하도록 하는 점 $P(a,b)$를 모두 찾을 때, $a+b$의 최댓값은?

<!-- Tick marks and labels on axes -->
<g class="label" transform="scale(0.0333, -0.0333)">
    <text x="-5" y="10" text-anchor="end">O</text>
    <text x="30" y="10">1</text>
    <text x="60" y="10">2</text>
    <text x="90" y="10">3</text>
    <text x="120" y="10">4</text>
    <text x="150" y="10">5</text>
    <text x="0" y="-35" text-anchor="middle">1</text>
    <text x="0" y="-65" text-anchor="middle">2</text>
    <text x="0" y="65" text-anchor="middle">-2</text>
    <text x="0" y="35" text-anchor="middle">-1</text>
</g>

<!-- Circles -->
<circle cx="0" cy="0" r="1" class="circle"/>
<text x="-0.3" y="0.5" class="label-purple" transform="scale(0.0333, -0.0333)">C₁: x²+y²=1</text>
<circle cx="3" cy="0" r="2" class="circle"/>
<text x="2.7" y="1.5" class="label-purple" transform="scale(0.0333, -0.0333)">C₂: (x-3)²+y²=4</text>

<!-- Centers -->
<circle cx="0" cy="0" r="0.05" class="point"/>
<text x="-0.3" y="-0.3" class="label" transform="scale(0.0333, -0.0333)">O₁</text>
<circle cx="3" cy="0" r="0.05" class="point"/>
<text x="3.1" y="-0.3" class="label" transform="scale(0.0333, -0.0333)">O₂</text>

<!-- Line L1: y=1 (one of the two lines L passing through P) -->
<line x1="-1.5" y1="1" x2="4.5" y2="1" class="main-line"/>
<text x="4.6" y="1" class="label" transform="scale(0.0333, -0.0333)">L₁</text>

<!-- Tangency condition for L1 on C1 -->
<circle cx="0" cy="1" r="0.05" class="bold-point"/>
<line x1="0" y1="0" x2="0" y2="1" class="dashed-line"/>
<text x="-0.1" y="0.7" class="label-purple" transform="scale(0.0333, -0.0333)">r₁=1</text>

<!-- Chord condition for L1 on C2 -->
<!-- Intersection points (3-sqrt(3), 1) and (3+sqrt(3), 1) -->
<circle cx="1.268" cy="1" r="0.05" class="bold-point"/>
<circle cx="4.732" cy="1" r="0.05" class="bold-point"/>
<line x1="1.268" y1="1" x2="4.732" y2="1" class="line" stroke-width="3" stroke-linecap="round"/>
<text x="2.5" y="1.3" class="label" transform="scale(0.0333, -0.0333)">A</text>
<text x="4" y="1.3" class="label" transform="scale(0.0333, -0.0333)">B</text>

<!-- Perpendicular from O2 to L1 (midpoint of AB) -->
<line x1="3" y1="0" x2="3" y2="1" class="dashed-line"/>
<circle cx="3" cy="1" r="0.05" class="point"/>
<text x="3.1" y="0.7" class="label-purple" transform="scale(0.0333, -0.0333)">d(O₂,L₁)=1</text>
<line x1="3" y1="0" x2="1.268" y2="1" class="dashed-line"/>
<text x="2.1" y="0.3" class="label-purple" transform="scale(0.0333, -0.0333)">r₂=2</text>
<text x="3.5" y="1.1" class="label-purple" transform="scale(0.0333, -0.0333)">AB = 2√3</text>

<!-- Point P: (a,b) for maximum a+b, which is ((3+sqrt(5))/2, 1) -->
<circle cx="2.61803" cy="1" r="0.08" class="bold-point"/>
<text x="2.61803" y="1.2" class="label" transform="scale(0.0333, -0.0333)">P</text>

<!-- The second line L (L4: 2x - sqrt(5)y - 3 = 0) passing through P -->
<!-- Drawn from x=0 to x=3.5 -->
<line x1="0" y1="-1.342" x2="3.5" y2="1.78885" class="line"/>
<text x="0.1" y="-1.2" class="label" transform="scale(0.0333, -0.0333)">L₄</text>

<!-- Tangency condition for L4 on C1 -->
<!-- Tangent point at (2/3, -sqrt(5)/3) -->
<circle cx="0.66667" cy="-0.745" r="0.05" class="bold-point"/>
<line x1="0" y1="0" x2="0.66667" y2="-0.745" class="dashed-line"/>
<text x="0.6" y="-0.5" class="label-purple" transform="scale(0.0333, -0.0333)">r₁=1</text>