{"id":9102,"date":"2026-02-21T19:15:22","date_gmt":"2026-02-21T13:45:22","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=9102"},"modified":"2026-02-21T19:15:31","modified_gmt":"2026-02-21T13:45:31","slug":"straight-lines-short-notes-formula-for-jee-main-and-advanced","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/straight-lines-short-notes-formula-for-jee-main-and-advanced\/","title":{"rendered":"Straight Lines Short Notes &#038; Formula for JEE Main  and Advanced"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Distance Formula<\/h2>\n\n\n\n<p>For two points \\( P(x_1, y_1) \\) and \\( Q(x_2, y_2) \\) in the Cartesian plane, the distance between them is given by:<br><br>\\[<br>D = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}<br>\\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Section Formula<\/h2>\n\n\n\n<p>For two points \\( A(x_1, y_1) \\) and \\( B(x_2, y_2) \\), the coordinates ( (x, y) ) of a point ( P ) dividing the line segment ( AB ) internally in the ratio ( m:n ) are:<br><br>\\[<br>x = \\frac{m x_2 + n x_1}{m + n}<br>\\]<br><br>\\[<br>y = \\frac{m y_2 + n y_1}{m + n}<br>\\]<\/p>\n\n\n\n<p>For two points \\( A(x_1, y_1) \\) and \\( B(x_2, y_2) \\), the coordinates ( (x, y) ) of a point ( P ) dividing the line segment ( AB ) externally in the ratio ( m:n ) are:<\/p>\n\n\n\n<p>\\[<br>x = \\frac{m x_2 &#8211; n x_1}{m &#8211; n}<br>\\]<br><br>\\[<br>y = \\frac{m y_2 &#8211; n y_1}{m &#8211; n}<br>\\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Mid Point Formula<\/h2>\n\n\n\n<p>For two points \\( A(x_1, y_1) \\) and \\( B(x_2, y_2) \\), the coordinates ( (x, y) ) of a mid point ( P ) is :<br><br>\\[<br>x = \\frac{x_2 + x_1}{2}<br>\\]<br><br>\\[<br>y = \\frac{y_2 + y_1}{2}<br>\\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Area of Triangle<\/h2>\n\n\n\n<p>Area of triangle ABC of coordinates \\( A(x_1, y_1) \\) , \\( B(x_2, y_2) \\) and (\\ C(x_3, y_3) \\)<br><br>$A=\\frac {1}{2}|x_1 (y_2-y_3 )+x_2 (y_3-y_1 )+x_3 (y_1-y_2 )|$<br><br><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Formula For Collinearity<\/h2>\n\n\n\n<p>For point A, B and C to be collinear, the value of A should be zero<br><br>$|x_1 (y_2-y_3 )+x_2 (y_3-y_1 )+x_3 (y_1-y_2 )|=0$<br><br><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Definition of Slope of the Straight line<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If $\\theta$ is the inclination of a line l, then $\\tan \\theta$ is called the slope or gradient of the line l.<\/li>\n\n\n\n<li>The slope of a line whose inclination is 90\u00b0 is not defined.<\/li>\n\n\n\n<li>The slope of a line is denoted by m.<\/li>\n\n\n\n<li>Thus, $m = \\tan \\theta$, $\\theta \\ne 90$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Slope Formula<\/h2>\n\n\n\n<p>\\[ m = \\frac{y_2 &#8211; y_1}{x_2 &#8211; x_1} \\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Angle Between Two Lines<\/h2>\n\n\n\n<p>If \\( m_1 \\) and \\( m_2 \\) are the slopes of two lines, then the angle ( $\\theta$ ) between them is:<\/p>\n\n\n\n<p>\\[<br>\\tan \\theta =| \\frac{m_2 &#8211; m_1}{1 + m_1 m_2}|<br>\\]<\/p>\n\n\n\n<p>If the lines are parallel, then $m_1=m_2$<\/p>\n\n\n\n<p>if the lines are perpendicular $m_1m_2 =-1$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">General Equation of Straight line<\/h2>\n\n\n\n<p>An equation of the form 0, $ax+ by+ c=0$ where  a, b, c are constants and  a, b are not simultaneously zero, always represents a straight line.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Various form Equation of Straight line<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Point-slope form<\/h3>\n\n\n\n<p>For a line with slope ( m ) passing through the point \\((x_1, y_1) \\)<\/p>\n\n\n\n<p>\\[<br>y &#8211; y_1 = m(x &#8211; x_1)<br>\\]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Two-point form<\/h3>\n\n\n\n<p>For a line passing through two points \\( (x_1, y_1) \\) and \\( (x_2, y_2) \\)<\/p>\n\n\n\n<p>\\[<br>y &#8211; y_1 = \\frac{y_2 &#8211; y_1}{x_2 &#8211; x_1} (x &#8211; x_1)<br>\\]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Slope Intercept Form<\/h3>\n\n\n\n<p>For a line with slope ( m )  and which cuts an intercept c on the y-axis<\/p>\n\n\n\n<p>$y=mx +c $<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Intercept Form<\/h3>\n\n\n\n<p>For a line intercepting the x-axis at ( a ) and the y-axis at ( b )<\/p>\n\n\n\n<p>\\[<br>\\frac{x}{a} + \\frac{y}{b} = 1<br>\\]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Normal Form<\/h3>\n\n\n\n<p>For a line making an angle \\( \\omega \\) with the positive x-axis and having a perpendicular distance ( p ) from the origin:<\/p>\n\n\n\n<p>\\[<br>x \\cos \\omega + y \\sin \\omega = p<br>\\]<\/p>\n\n\n\n<p>In normal form of equation of a straight line p is always taken as positive and ? is measured from positive direction of x-axis in anticlockwise direction between 0 and $2\\pi$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Parametric Form<\/h3>\n\n\n\n<p>The equation of a straight line passing through the point $(x_1 ,y_1)$ and making an angle $\\theta$ with the positive direction of x-axis is<\/p>\n\n\n\n<p>$\\frac {x-x_1}{\\cos \\theta}= \\frac {y-y_1}{\\sin \\theta}= r$<\/p>\n\n\n\n<p>Here $0 \\leq \\theta \\leq \\pi$<\/p>\n\n\n\n<p>r is the distance of the point (x,y) from $(x_1,y_2)$<\/p>\n\n\n\n<p>The coordinates of any point on the line at a distance r from the point $(x_1, y_1)$ can be taken as<br>$(x_1 + r cos \\theta, y_1 + r sin \\theta)$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Equations  of straight lines passing through a given point and making an given angle with a given line<\/h2>\n\n\n\n<p>Equation of line passing through the point $(x_1,y_1)$ and making an angle $\\alpha $ with the given line $y=m_1 x + c$ is given as<\/p>\n\n\n\n<p>$y-y_1= \\frac {m_1 \\pm tan \\alpha}{1 \\mp m_1 tan \\alpha} (x-x_1)$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Length  of Perpendicular Point From a Line<\/h2>\n\n\n\n<p>The distance of a point from a line is the length of the perpendicular drawn from the point to the line. Let L : Ax + By + C = 0 be a line, whose distance from the point P (x1, y1) is d. Then d is given by<br><br>$d= \\frac {|Ax_1 + By+1 + C|}{\\sqrt {A^2 + B^2}}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Distance between two Parallel Lines<\/h2>\n\n\n\n<p>For two parallel lines <br>$y=mx + c_1$<br>$y=mx+ c_2$<br>Distance between them is given by<br>$d= \\frac {|c_1 -c_2|}{\\sqrt {1+ m^2}}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Position of two points with respect to given line<\/h2>\n\n\n\n<p>For points $(x_1,y_1)$ and $(x_2,y_2)$ with respect to given line $ax+ By + c =0$<\/p>\n\n\n\n<p>if $\\frac {ax_1 + by_1 + c}{ax_2 + by_2 + c} &lt; 0$<\/p>\n\n\n\n<p>Then points are opposite side of the line<\/p>\n\n\n\n<p>if $\\frac {ax_1 + by_1 + c}{ax_2 + by_2 + c} &gt; 0$<\/p>\n\n\n\n<p>Then the points are on same side<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Family of Lines passing through intersection of two lines<\/h2>\n\n\n\n<p>if $L=0$ and $L^{&#8216;}=0$ be the two line, then family of lines passing through the intersection of two lines is <\/p>\n\n\n\n<p>$L+ \\lambda L^{&#8216;}=0$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Concurrency of Straight lines<\/h2>\n\n\n\n<p>For the three lines to be concurrent<\/p>\n\n\n\n<p>$a_1 x + b_1 y + c_1=0$<br>$a_2 x + b_2 y + c_2=0$<br>$a_3 x + b_13y + c_3=0$<\/p>\n\n\n\n<p>we should have<br>$ \\begin{vmatrix}<br>a_1 &amp; b_1 &amp; c_1 \\\\<br>a_2 &amp; b_2 &amp; c_2 \\\\<br>a_3 &amp; b_3 &amp; c_3<br>\\end{vmatrix}<br>$<\/p>\n\n\n\n<p>or<\/p>\n\n\n\n<p>$mL_1 + nL_2+ kl_3=0$ where m,n and k are constant which are non zero at the same time<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Equations of the Bisectors of the angles between the lines<\/h2>\n\n\n\n<p>The equations of the bisectors of the angles between the lines $a_1 x + b_1 y + c_1=0$ and $a_2 x + b_2 y + c_2=0$ is<\/p>\n\n\n\n<p>$\\frac {a_1 x + b_1 y +c_1}{\\sqrt {a_1^2 + b_1^2} }=\\pm \\frac {a_2 x + b_2 y +c_2}{\\sqrt {a_2^2 + b_2^2} }$<\/p>\n\n\n\n<p><strong>How to Find the equation of bisector of obtuse  and acute angle<\/strong><\/p>\n\n\n\n<p>Let the line be such that $a_1 x + b_1 y + c_1=0$ and $a_2 x + b_2 y + c_2=0$  and $c_1 &gt;0$ and $c_2&gt; 0$<\/p>\n\n\n\n<p>Then the equation of bisector containing Origin<\/p>\n\n\n\n<p>$\\frac {a_1 x + b_1 y +c_1}{\\sqrt {a_1^2 + b_1^2} }=   \\frac {a_2 x + b_2 y +c_2}{\\sqrt {a_2^2 + b_2^2} }$ <\/p>\n\n\n\n<p>And the other line is<\/p>\n\n\n\n<p>$\\frac {a_1 x + b_1 y +c_1}{\\sqrt {a_1^2 + b_1^2} }=  &#8211; \\frac {a_2 x + b_2 y +c_2}{\\sqrt {a_2^2 + b_2^2} }$ <\/p>\n\n\n\n<p>Now<\/p>\n\n\n\n<p>If $a_1a_2 + b_1b_2 &gt;0$  then the origin lies in obtuse angle i.e., the bisector containing origin is obtuse angle bisector and if $a_1a_2 + b_1b_2 &lt;0$ then the origin lies in acute angle i.e., the bisector containing origin is acute angle bisector.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"> Foot of The Perpendicular from a point on Line<\/h2>\n\n\n\n<p>The co-ordinate of foot of perpendicular $x_2 ,y_2$ of a given point $x_1,y_1$ on the line $ax+ by+ c=0$ are given by<\/p>\n\n\n\n<p>$\\frac {x_2 -x_1}{a} = \\frac {y_2 -y_1}{b} = &#8211; \\frac {ax_1 + by_1 + c}{a^2 + b^2}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Image of a point on a Line<\/h2>\n\n\n\n<p>The co-ordinate of image  $x_2 ,y_2$ of a given point $x_1,y_1$ on the line $ax+ by+ c=0$ are given by<\/p>\n\n\n\n<p>$\\frac {x_2 -x_1}{a} = \\frac {y_2 -y_1}{b} = -2 \\frac {ax_1 + by_1 + c}{a^2 + b^2}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Standard Points of the Triangle<\/h2>\n\n\n\n<p><strong>Centroid Of the Triangle<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It is the concurrent point of the Median of the triangle<\/li>\n\n\n\n<li>Centroid divides the median in the ratio  2: 1<\/li>\n\n\n\n<li>Coordinates are given by<\/li>\n<\/ul>\n\n\n\n<p>$x= \\frac {x_1 + x_2 + x_3}{3}$ and $y= \\frac {y_1+ y_2 + y_3}{3}$<\/p>\n\n\n\n<p><strong>Incenter of the Triangle<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It is the concurrent point of the angle bisectors<\/li>\n\n\n\n<li>Coordinates are given by<\/li>\n<\/ul>\n\n\n\n<p>$x=\\frac {ax_1 + bx_2 + cx_3}{a+ b+ c}$ and $y=\\frac {ay_1 + by_2 + cy_3}{a+ b+c}$<\/p>\n\n\n\n<p>Here a, b, c are the sides of the triangles<\/p>\n\n\n\n<p><strong>Circumcenter of the Triangle<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It is the concurrent point of the perpendicular bisectors of the sides of the triangles.<\/li>\n\n\n\n<li>If A, B, C are the angles of the $\\Delta ABC$ and Vertices are \\( A(x_1, y_1) \\) , \\( B(x_2, y_2) \\) and \\(C(x_3, y_3) \\), then circumcenter point is given as<\/li>\n<\/ul>\n\n\n\n<p>$x=\\frac {x_1 sin2A + x_2 sin2B + x_3 sin2C}{sin2A + sin2B+ sin2C}$ and $y=\\frac {y_1 sin2A + y_2 sin2B+ y_3 sin2C}{sin2A + sin2B+ sin2C}$<\/p>\n\n\n\n<p><strong>Orthocenter of the Triangle<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The orthocenter of a triangle is the point of intersection of altitudes<\/li>\n\n\n\n<li>For an acute triangle, it lies inside the triangle.<\/li>\n\n\n\n<li>For an obtuse triangle, it lies outside of the triangle.<\/li>\n\n\n\n<li>If A, B, C are the angles of the $\\Delta ABC$ and Vertices are \\( A(x_1, y_1) \\) , \\( B(x_2, y_2) \\) and \\(C(x_3, y_3) \\), then orthocenter point is given as<\/li>\n<\/ul>\n\n\n\n<p>$x=\\frac {x_1  Tan A + x_2 Tan B + x_3 Tan C}{Tan A + Tan B+  Tan C}$ and $y=\\frac {y_1 Tan A + y_2  Tan B+ y_3  Tan C}{Tan A + Tan B+  Tan C}$<\/p>\n\n\n\n<p><strong>Important Points<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>In right angled  the right angled vertex is the orthocenter of  triangle and mid-point of hypotenuse is circumcenter<\/li>\n\n\n\n<li>In a equilateral Triangle, Centroid, Incenter, Circumcenter and Orthocenter coincides<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Distance Formula For two points \\( P(x_1, y_1) \\) and \\( Q(x_2, y_2) \\) in the Cartesian plane, the distance between them is given by: \\[D = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\\] Section Formula For two points \\( A(x_1, y_1) \\) and \\( B(x_2, y_2) \\), the coordinates ( (x, y) ) of a point ( P [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","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":"","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":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-9102","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Straight Lines Short Notes &amp; Formula for JEE Main and Advanced - physicscatalyst&#039;s Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/straight-lines-short-notes-formula-for-jee-main-and-advanced\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Straight 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