{"id":5226,"date":"2025-08-02T20:29:00","date_gmt":"2025-08-03T00:29:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=5226"},"modified":"2025-08-30T20:30:32","modified_gmt":"2025-08-31T00:30:32","slug":"hyperbolas-covered-by-a-quadrilateral","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=5226","title":{"rendered":"Hyperbolas Covered by a Quadrilateral"},"content":{"rendered":"\n<p>Four points $A$, $B$, $C$ and $D$ are chosen on each of $4$ hyperbola branches of $x^2y^2=1$ (as\u00a0$y=\\dfrac{1}{x}$\u00a0and\u00a0$y=-\\dfrac{1}{x}$\u00a0combined). Find the minimum area of the quadrilateral.<a href=\"javascript:toggle_visibility('hyperbola_chall_2025_08_02')\">\ud83d\udd11<\/a><\/p>\n<div id=\"hyperbola_chall_2025_08_02\" style=\"display:none\"><\/p>\n\n\n\n<p><strong>Solution<\/strong>: Given that all 4 vertices of the quadrilaterial are on the $4$ branches, let $A=(a,\\dfrac{1}{a})$, $B=(-b,\\dfrac{1}{b})$, $C=(-c,-\\dfrac{1}{c})$, and $D=(d,-\\dfrac{1}{d})$, where $a,b,c,d>0$. <\/p>\n\n\n\n<p>By the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Shoelace_formula\">shoelace formula<\/a>, we have <\/p>\n\n\n\n<p>$$[ABCD]=\\dfrac{1}{2}\\Big{(}(a\\cdot\\dfrac{1}{b}-\\dfrac{1}{a}\\cdot(-b))+((-b)\\cdot(-\\dfrac{1}{c})-\\dfrac{1}{b}\\cdot(-c))$$<\/p>\n\n\n\n<p>$$+(-c\\cdot(-\\dfrac{1}{d})-(-\\dfrac{1}{c}\\cdot d))+(d\\cdot\\dfrac{1}{a}-a\\cdot(-\\dfrac{1}{d}))\\Big{)}\\tag{1}$$<\/p>\n\n\n\n<p>$$=\\dfrac{1}{2}\\Big{(}(\\dfrac{a}{b}+\\dfrac{b}{a})+(\\dfrac{b}{c}+\\dfrac{c}{b})+(\\dfrac{c}{d}+\\dfrac{d}{c})+(\\dfrac{d}{a}+\\dfrac{a}{d})\\Big{)}$$<\/p>\n\n\n\n<p>By <a href=\"https:\/\/en.wikipedia.org\/wiki\/AM\u2013GM_inequality\">AM\u2013GM inequality<\/a>, for $x,y>0$ we have $$\\dfrac{x}{y}+\\dfrac{y}{x}\\ge2\\sqrt{\\dfrac{x}{y}\\cdot\\dfrac{y}{x}}=2\\tag{2}$$<\/p>\n\n\n\n<p>Applying inequality $(2)$ to equation $(1)$, we have $$[ABCD]\\ge\\dfrac{1}{2}\\Big{(}(2+2+2+2)\\Big{)}=4$$<\/p>\n\n\n\n<p>Therefore, the minimum area of the quadrilateral is $\\boxed{4}$.<\/p>\n\n\n\n<p>Note: It can be proved that the smallest area of any quadrilaterals intersecting all $4$ hyperbola branches of $x^2y^2=1$ is $4$. Combining the answer to this <a href=\"\/mlog\/?p=5130\">question<\/a>, we can prove that the smallest area of any convex polygons intersecting all $4$ hyperbola branches of $x^2y^2=1$ is $4$, as any convex polygons with more than $4$ sides can be reduced to a smaller quadrilateral by removing a vertex.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Four points $A$, $B$, $C$ and $D$ are chosen on each of $4$ hyperbola branches of $x^2y^2=1$ (as\u00a0$y=\\dfrac{1}{x}$\u00a0and\u00a0$y=-\\dfrac{1}{x}$\u00a0combined). Find the minimum area of the quadrilateral.\ud83d\udd11 Solution: Given that all 4 vertices of the quadrilaterial are on the $4$ branches, let &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=5226\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13,17,9],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5226"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5226"}],"version-history":[{"count":19,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5226\/revisions"}],"predecessor-version":[{"id":5246,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5226\/revisions\/5246"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5226"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5226"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5226"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}