{"id":192897,"date":"2025-02-18T10:22:39","date_gmt":"2025-02-18T10:22:39","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=192897"},"modified":"2025-02-18T10:22:41","modified_gmt":"2025-02-18T10:22:41","slug":"test-the-polar-equation-for-symmetry-with-respect-to-the-polar-axis","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/18\/test-the-polar-equation-for-symmetry-with-respect-to-the-polar-axis\/","title":{"rendered":"Test the polar equation for symmetry with respect to the polar axis"},"content":{"rendered":"\n<p>Test the polar equation for symmetry with respect to the polar axis, the pole, and the line 0 = 1 (Select all that apply.) 2 r= 9 sec(O) O symmetric with respect to the polar axis symmetric with respect to the pole O symmetric with respect to the line 0 = 1 7 not symmetric with respect to any of these &#8211; [-12.12 Points] DETAILS SPRECALC7 8.2.015. Test the polar equation for symmetry with respect to the polar axis, the pole, and the line 0 = 1. (Select all that apply.) p2 = 16 cos(20) symmetric with respect to the polar axis O symmetric with respect to the pole O symmetric with respect to the line 0 = 1 7 O not symmetric with respect to any of these<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s analyze the given polar equations for symmetry with respect to the polar axis, the pole, and the line ( \\theta = 1 ).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Polar Equation 1: ( r = 9 \\sec(\\theta) )<\/h3>\n\n\n\n<p><strong>Symmetry with respect to the polar axis:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For symmetry with respect to the polar axis, we check if replacing ( \\theta ) with ( -\\theta ) results in the same equation.<\/li>\n\n\n\n<li>The equation becomes:<br>[<br>r = 9 \\sec(-\\theta)<br>]<br>Since ( \\sec(-\\theta) = \\sec(\\theta) ), the equation remains the same.<\/li>\n\n\n\n<li>Therefore, <strong>this equation is symmetric with respect to the polar axis<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p><strong>Symmetry with respect to the pole:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For symmetry with respect to the pole, we check if replacing ( r ) with ( -r ) results in the same equation.<\/li>\n\n\n\n<li>The equation becomes:<br>[<br>-r = 9 \\sec(\\theta)<br>]<br>This equation is not the same as the original, so it does <strong>not<\/strong> have symmetry with respect to the pole.<\/li>\n<\/ul>\n\n\n\n<p><strong>Symmetry with respect to the line ( \\theta = 1 ):<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>To check for symmetry with respect to the line ( \\theta = 1 ), we would typically replace ( \\theta ) with ( 2 &#8211; \\theta ) and see if the equation remains unchanged.<\/li>\n\n\n\n<li>Substituting into the equation:<br>[<br>r = 9 \\sec(2 &#8211; \\theta)<br>]<br>This does not simplify to the original equation, so it is <strong>not symmetric with respect to the line ( \\theta = 1 )<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Polar Equation 2: ( r^2 = 16 \\cos(2\\theta) )<\/h3>\n\n\n\n<p><strong>Symmetry with respect to the polar axis:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For symmetry with respect to the polar axis, we replace ( \\theta ) with ( -\\theta ) and check if the equation stays the same:<br>[<br>r^2 = 16 \\cos(-2\\theta) = 16 \\cos(2\\theta)<br>]<br>Since the equation remains unchanged, <strong>this equation is symmetric with respect to the polar axis<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p><strong>Symmetry with respect to the pole:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For symmetry with respect to the pole, we replace ( r ) with ( -r ) and check if the equation remains the same:<br>[<br>(-r)^2 = 16 \\cos(2\\theta)<br>]<br>Since ( (-r)^2 = r^2 ), the equation remains unchanged. Therefore, <strong>this equation is symmetric with respect to the pole<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p><strong>Symmetry with respect to the line ( \\theta = 1 ):<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Substituting ( 2 &#8211; \\theta ) for ( \\theta ):<br>[<br>r^2 = 16 \\cos(2(2 &#8211; \\theta)) = 16 \\cos(4 &#8211; 2\\theta)<br>]<br>This does <strong>not<\/strong> simplify to the original equation, so the equation is <strong>not symmetric with respect to the line ( \\theta = 1 )<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For the first equation ( r = 9 \\sec(\\theta) ), it is <strong>symmetric with respect to the polar axis<\/strong>, but <strong>not symmetric with respect to the pole or the line ( \\theta = 1 )<\/strong>.<\/li>\n\n\n\n<li>For the second equation ( r^2 = 16 \\cos(2\\theta) ), it is <strong>symmetric with respect to the polar axis and the pole<\/strong>, but <strong>not symmetric with respect to the line ( \\theta = 1 )<\/strong>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Test the polar equation for symmetry with respect to the polar axis, the pole, and the line 0 = 1 (Select all that apply.) 2 r= 9 sec(O) O symmetric with respect to the polar axis symmetric with respect to the pole O symmetric with respect to the line 0 = 1 7 not symmetric [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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