{"id":1995,"date":"2023-07-04T18:57:00","date_gmt":"2023-07-04T22:57:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=1995"},"modified":"2024-10-25T12:04:34","modified_gmt":"2024-10-25T16:04:34","slug":"algebra-challenge-2-%e2%ad%90%e2%ad%90","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=1995","title":{"rendered":"Algebra Challenge &#8211; 2 \u2b50\u2b50"},"content":{"rendered":"\n<p>On the $X$-$Y$ plane, two circles centered at $(0,0)$ with radius $1$ and $2$ respectively. Let point $A=(-1,0)$, $B=(1,0)$, and $C$ is a point on the bigger circle. Find the locus of the orthocenter $P$ of $\\triangle{ABC}$. Click <a href=\"javascript:toggle_visibility(&quot;alg-chall-2-sol&quot;);\">here<\/a> for the solution.<\/p>\n\n\n\n<div id=\"alg-chall-2-sol\" style=\"display:none\">\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" src=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2021\/07\/Screen-Shot-2021-07-26-at-2.45.11-PM-935x1024.png\" alt=\"\" class=\"wp-image-2001\" width=\"234\" height=\"256\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2021\/07\/Screen-Shot-2021-07-26-at-2.45.11-PM-935x1024.png 935w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2021\/07\/Screen-Shot-2021-07-26-at-2.45.11-PM-274x300.png 274w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2021\/07\/Screen-Shot-2021-07-26-at-2.45.11-PM-768x841.png 768w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2021\/07\/Screen-Shot-2021-07-26-at-2.45.11-PM.png 1096w\" sizes=\"(max-width: 234px) 100vw, 234px\" \/><\/figure>\n\n\n\n<p><strong>Solution<\/strong>: The orthocenter $P$ is the point where line $AP$ and $CP$ intersects with either other. Assume $C=(X_c, Y_c)$, we have $X_c^2+Y_c^2=4$. Therefore $$Y_c=\\pm\\sqrt{4-X_c^2}\\tag{1}$$ And the equation of line $CP$ is $$x=X_c\\tag{2}$$ Because the slop of $BC$ is $\\dfrac{Y_c}{X_c-1}$, the equation of line $AP$ is $$y=-\\dfrac{X_c-1}{Y_c}(x+1)\\tag{3}$$ By substituting $X_c$ and $Y_c$ in equation (3) with $(1)$ and $(2)$, we have $$y=\\dfrac{1-x^2}{\\pm\\sqrt{4-x^2}}$$ Squaring both side of the above equation, the locus of point $P$ is the following function: $$\\boxed{y^2=\\dfrac{(1-x^2)^2}{4-x^2}}$$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>On the $X$-$Y$ plane, two circles centered at $(0,0)$ with radius $1$ and $2$ respectively. Let point $A=(-1,0)$, $B=(1,0)$, and $C$ is a point on the bigger circle. Find the locus of the orthocenter $P$ of $\\triangle{ABC}$. Click here for &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=1995\">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":[13,14],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/1995"}],"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=1995"}],"version-history":[{"count":15,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/1995\/revisions"}],"predecessor-version":[{"id":2137,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/1995\/revisions\/2137"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1995"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1995"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1995"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}