{"status": "success", "data": {"description_md": "Circles with centers $(2,4)$ and $(14,9)$ have radii $4$ and $9$, respectively. The equation of a common external tangent to the circles can be written in the form $y=mx+b$ with $m>0$. What is $b$? <br><center><img class=\"problem-image\" alt=\"[asy] size(150); defaultpen(linewidth(0.7)+fontsize(8)); draw(circle((2,4),4));draw(circle((14,9),9)); draw((0,-2)--(0,20));draw((-6,0)--(25,0)); draw((2,4)--(2,4)+4*expi(pi*4.5/11)); draw((14,9)--(14,9)+9*expi(pi*6/7)); label(&quot;4&quot;,(2,4)+2*expi(pi*4.5/11),(-1,0)); label(&quot;9&quot;,(14,9)+4.5*expi(pi*6/7),(1,1)); label(&quot;(2,4)&quot;,(2,4),(0.5,-1.5));label(&quot;(14,9)&quot;,(14,9),(1,-1)); draw((-4,120*-4/119+912/119)--(11,120*11/119+912/119)); dot((2,4)^^(14,9)); [/asy]\" class=\"latexcenter\" height=\"178\" src=\"https://latex.artofproblemsolving.com/e/e/c/eec9e573082823c38ceb230bca0dbe46393fdbfe.png\" width=\"252\"/></center>\n\n$\\mathrm{(A) \\ } \\frac{908}{119}\\qquad \\mathrm{(B) \\ } \\frac{909}{119}\\qquad \\mathrm{(C) \\ } \\frac{130}{17}\\qquad \\mathrm{(D) \\ } \\frac{911}{119}\\qquad \\mathrm{(E) \\ }  \\frac{912}{119}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Circles with centers  <span class=\"katex--inline\">(2,4)</span>  and  <span class=\"katex--inline\">(14,9)</span>  have radii  <span class=\"katex--inline\">4</span>  and  <span class=\"katex--inline\">9</span> , respectively. The equation of a common external tangent to the circles can be written in the form  <span class=\"katex--inline\">y=mx+b</span>  with  <span class=\"katex--inline\">m&gt;0</span> . What is  <span class=\"katex--inline\">b</span> ? <br/><center><img class=\"latexcenter\" alt=\"[asy] size(150); defaultpen(linewidth(0.7)+fontsize(8)); draw(circle((2,4),4));draw(circle((14,9),9)); draw((0,-2)--(0,20));draw((-6,0)--(25,0)); draw((2,4)--(2,4)+4*expi(pi*4.5/11)); draw((14,9)--(14,9)+9*expi(pi*6/7)); label(&#34;4&#34;,(2,4)+2*expi(pi*4.5/11),(-1,0)); label(&#34;9&#34;,(14,9)+4.5*expi(pi*6/7),(1,1)); label(&#34;(2,4)&#34;,(2,4),(0.5,-1.5));label(&#34;(14,9)&#34;,(14,9),(1,-1)); draw((-4,120*-4/119+912/119)--(11,120*11/119+912/119)); dot((2,4)^^(14,9)); [/asy]\" height=\"178\" src=\"https://latex.artofproblemsolving.com/e/e/c/eec9e573082823c38ceb230bca0dbe46393fdbfe.png\" width=\"252\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A) \\ } \\frac{908}{119}\\qquad \\mathrm{(B) \\ } \\frac{909}{119}\\qquad \\mathrm{(C) \\ } \\frac{130}{17}\\qquad \\mathrm{(D) \\ } \\frac{911}{119}\\qquad \\mathrm{(E) \\ }  \\frac{912}{119}</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2006 AMC 12A Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/06_amc12A_p20", "prev": "/problem/06_amc12A_p18"}}