{"status": "success", "data": {"description_md": "The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\\frac yx$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$?\n\n$\\mathrm{(A)}\\ 3<br>\\qquad\\mathrm{(B)}\\ \\sqrt{10}<br>\\qquad\\mathrm{(C)}\\ \\frac 72<br>\\qquad\\mathrm{(D)}\\ \\frac 92<br>\\qquad\\mathrm{(E)}\\ 2\\sqrt{14}$\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>The graph of  <span class=\"katex--inline\">2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0</span>  is an ellipse in the first quadrant of the  <span class=\"katex--inline\">xy</span> -plane. Let  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  be the maximum and minimum values of  <span class=\"katex--inline\">\\frac yx</span>  over all points  <span class=\"katex--inline\">(x,y)</span>  on the ellipse. What is the value of  <span class=\"katex--inline\">a+b</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A)}\\ 3\\qquad\\mathrm{(B)}\\ \\sqrt{10}\\qquad\\mathrm{(C)}\\ \\frac 72\\qquad\\mathrm{(D)}\\ \\frac 92\\qquad\\mathrm{(E)}\\ 2\\sqrt{14}</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": 5, "problem_name": "2004 AMC 12B Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/04_amc12B_p22", "prev": "/problem/04_amc12B_p20"}}