두 그래프 위의 점과 선분 길이를 이용한 미지수 찾기

문제

두 함수 $y = ax$와 $y = \frac{b}{x}$의 그래프가 있습니다. (단, $a$와 $b$는 자연수입니다.) 점 P는 함수 $y = ax$의 그래프 위의 점으로, $x$좌표는 3입니다. 점 Q는 함수 $y = \frac{b}{x}$의 그래프 위의 점으로, $x$좌표는 $k$입니다. (단, $k$는 자연수입니다.) 점 P와 Q는 모두 제1사분면에 있으며, $y$좌표가 같습니다. 또한, 선분 PQ의 길이는 5입니다. 이때, $a+b$의 최솟값은 무엇입니까?

<!-- Origin label -->
<text x="-0.8" y="-0.5" class="label" transform="scale(1, -1)">O</text>

<!-- Tick marks and labels for x-axis -->
<line x1="3" y1="0.1" x2="3" y2="-0.1" stroke="#1F2937" stroke-width="1"/>
<text x="3" y="-0.5" class="tick-label" transform="scale(1, -1)" text-anchor="middle">3</text>
<line x1="8" y1="0.1" x2="8" y2="-0.1" stroke="#1F2937" stroke-width="1"/>
<text x="8" y="-0.5" class="tick-label" transform="scale(1, -1)" text-anchor="middle">k</text>

<!-- Tick mark and label for y-axis -->
<line x1="0.1" y1="3" x2="-0.1" y2="3" stroke="#1F2937" stroke-width="1"/>
<text x="-0.2" y="3" class="tick-label" transform="scale(1, -1)" text-anchor="end" dy="0.3em">Y</text> 

<!-- Function graphs (values a=1, b=24 for illustrative minimum case) -->
<!-- y = ax (line passing through (0,0) and (3,3)) -->
<path d="M 0 0 L 4 4" class="graph-line" />
<text x="3.5" y="4.5" class="label" transform="scale(1, -1)" text-anchor="start">$y=ax$</text>

<!-- y = b/x (hyperbola passing through (8,3)) -->
<!-- Points: (3,8), (4,6), (5,4.8), (6,4), (7,3.43), (8,3), (9,2.67), (10,2.4) -->
<path d="M 3 8 L 4 6 L 5 4.8 L 6 4 L 7 3.43 L 8 3 L 9 2.67 L 10 2.4" class="graph-line" />
<text x="9" y="3.5" class="label" transform="scale(1, -1)" text-anchor="start">$y=\frac{b}{x}$</text>

<!-- Points P and Q -->
<circle cx="3" cy="3" r="0.1" class="point" />
<text x="3.2" y="3.2" class="label" transform="scale(1, -1)">P</text>

<circle cx="8" cy="3" r="0.1" class="point" />
<text x="8.2" y="3.2" class="label" transform="scale(1, -1)">Q</text>

<!-- Segment PQ and its length -->
<line x1="3" y1="3" x2="8" y2="3" class="dashed-line" />
<text x="5.5" y="3.5" class="label" transform="scale(1, -1)" text-anchor="middle">5</text>

<!-- Dashed lines from points to axes -->
<line x1="3" y1="3" x2="3" y2="0" class="dashed-line" />
<line x1="8" y1="3" x2="8" y2="0" class="dashed-line" />
<line x1="3" y1="3" x2="0" y2="3" class="dashed-line" />