{"status": "success", "data": {"description_md": "**POTD June 5, 2024**  \n  \nLet complex numbers $z$ and $\\dfrac{1}{z}$ lie on circles that are modeled by the following equations on the complex plane:  \n  \n**1.** $(x-\\alpha)^2 + (y - \\beta)^2 = r^2$  \n  \n**2**. $(x-\\alpha)^2 + (y - \\beta)^2 = 4r^2$  \n  \nIf $\\zeta = \\alpha + i\\beta$, and $2\\left|\\zeta\\right|^2 = r^2 + 2$, then $\\left|z\\right|$ can be expressed as $\\frac{\\sqrt{a}}{b}$. Compute $a+b$.", "description_html": "<p><strong>POTD June 5, 2024</strong></p>&#10;<p>Let complex numbers <span class=\"katex--inline\">z</span> and <span class=\"katex--inline\">\\dfrac{1}{z}</span> lie on circles that are modeled by the following equations on the complex plane:</p>&#10;<p><strong>1.</strong> <span class=\"katex--inline\">(x-\\alpha)^2 + (y - \\beta)^2 = r^2</span></p>&#10;<p><strong>2</strong>. <span class=\"katex--inline\">(x-\\alpha)^2 + (y - \\beta)^2 = 4r^2</span></p>&#10;<p>If <span class=\"katex--inline\">\\zeta = \\alpha + i\\beta</span>, and <span class=\"katex--inline\">2\\left|\\zeta\\right|^2 = r^2 + 2</span>, then <span class=\"katex--inline\">\\left|z\\right|</span> can be expressed as <span class=\"katex--inline\">\\frac{\\sqrt{a}}{b}</span>. Compute <span class=\"katex--inline\">a+b</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Problem of the Day #178", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}