{"id":5084,"date":"2012-12-01T02:03:31","date_gmt":"2012-12-01T02:03:31","guid":{"rendered":"http:\/\/schooltutoring.com\/scholarship\/?p=5084"},"modified":"2014-12-02T08:31:55","modified_gmt":"2014-12-02T08:31:55","slug":"angle-between-pair-of-straight-lines","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/scholarship\/2012\/12\/01\/angle-between-pair-of-straight-lines\/","title":{"rendered":"Angle Between Pair of Straight Lines"},"content":{"rendered":"<p><strong>Angle between pair of straight lines:<\/strong><\/p>\n<p>We know that the equation ax<sup>2<\/sup>+2hxy+by<sup>2<\/sup>=0 represents a pair of straight lines passing through origin and hence it can be written as product of two linear factors, ax<sup>2<\/sup>+2hxy+by<sup>2<\/sup>= (lx+my)(px+qy), where lp=a, mq=b and lq+mp=2h. Also, the separate equations of lines are lx+my=0 and px+qy=0. So, the angle between the lines is given by<\/p>\n<p>Cos \u03b8 = (lp+mq)\/\u221a[(l<sup>2<\/sup>+m<sup>2<\/sup>)(p<sup>2<\/sup>+q<sup>2<\/sup>)]\n<p>= = (lp+mq)\/\u221a[(lp-mq)<sup>2<\/sup> +(lq+mp)<sup>2<\/sup>]\n<p>= (a+b)\/\u221a[(a-b)<sup>2<\/sup> + 4h<sup>2<\/sup>]\n<p>The above formula for Cos \u03b8 gives the acute angle between the lines ax<sup>2<\/sup>+2hxy+by<sup>2<\/sup>=0 if a+b&gt;0 and the obtuse angle between the lines if a+b&lt;0. Hence the acute angle between the lines represented by ax<sup>2<\/sup>+2hxy+by<sup>2<\/sup>=0 is Cos<sup>-1<\/sup>[|a+b|\/\u221a[(a-b)<sup>2<\/sup> + 4h<sup>2<\/sup>]]\n<p>We know that ax<sup>2<\/sup>+2hxy+by<sup>2<\/sup>=0=0 represents a pair of parallel lines (coincident) if h<sup>2<\/sup>=ab. Now, the lines represented by ax<sup>2<\/sup>+2hxy+by<sup>2<\/sup>=0 are perpendicular if Cos \u03b8=0. i.e. the lines are perpendicular of a+b=0.<\/p>\n<p>(i.e. the lines are perpendicular if the sum of coefficients of x<sup>2<\/sup> and y<sup>2<\/sup> is zero).<\/p>\n<p>If a+b \u22600, then the lines represented by ax<sup>2<\/sup>+2hxy+by<sup>2<\/sup>=0 are not perpendicular nad the angle between them is represented by either of these formulas.<\/p>\n<p>Tan \u03b8 = [2\u221a(h2-ab)]\/(a+b)<\/p>\n<p>Cos \u03b8 = (a+b)\/\u221a[(a-b)<sup>2<\/sup> + 4h<sup>2<\/sup>]\n<p>Sin \u03b8 = [2\u221a(h2-ab)]\/\u221a[(a-b)<sup>2<\/sup> + 4h<sup>2<\/sup>]\n<p><span class=\"tutorOrange\">SchoolTutoring Academy<\/span> is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Georgia  visit: <a href=\"https:\/\/schooltutoring.com\/Georgia-Tutoring-Programs\/\">Tutoring in Georgia. <\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Angle between pair of straight lines: We know that the equation ax2+2hxy+by2=0 represents a pair of straight lines passing through origin and hence it can be written as product of two linear factors, ax2+2hxy+by2= (lx+my)(px+qy), where lp=a, mq=b and lq+mp=2h. Also, the separate equations of lines are lx+my=0 and px+qy=0. So, the angle between the [&hellip;]<\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[10],"tags":[94,414,1039,1274,1648,1805],"class_list":["post-5084","post","type-post","status-publish","format-standard","hentry","category-geometry","tag-angle","tag-cos","tag-line","tag-pair","tag-sin","tag-tan"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/5084","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/comments?post=5084"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/5084\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media?parent=5084"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/categories?post=5084"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/tags?post=5084"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}