{"id":3892,"date":"2021-08-09T20:48:49","date_gmt":"2021-08-09T20:48:49","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3892"},"modified":"2021-11-30T16:40:07","modified_gmt":"2021-11-30T11:10:07","slug":"formula-for-angle-between-two-lines","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/formula-for-angle-between-two-lines\/","title":{"rendered":"Formula for Angle between Two Lines"},"content":{"rendered":"
Here, you will learn formula for angle between two lines, equation of straight line making an angle with a given line and reflection and image of a point in a line and also length of perpendicular from a point on a line.<\/p>\n
Let’s begin –<\/p>\n
(a)\u00a0 If \\(\\theta\\) be the angle between two lines : y = \\(m_1x + c_1\\) and y = \\(m_2x + c_2\\), then<\/p>\n
\ntan\\(\\theta\\) = \\(\\pm\\) (\\(m_1-m_2\\over {1+m_1m_2}\\))<\/p>\n<\/blockquote>\n
(b)\u00a0 If the equation of lines are \\(a_1x + b_1y + c_1\\) = 0 and \\(a_2x + b_2y + c_2\\) = 0, then these lines are –\u00a0<\/p>\n
\n(i)\u00a0 Parallel\u00a0 \\(\\iff\\)\u00a0 \u00a0\\(a_1\\over a_2\\) = \\(b_1\\over b_2\\) \\(\\ne\\) \\(c_1\\over c_2\\)<\/p>\n
(ii)\u00a0 Perpendicular\u00a0 \\(\\iff\\)\u00a0 \\(a_1a_2\\) + \\(b_1b_2\\) = 0<\/p>\n
(iii)\u00a0 Coincident\u00a0 \\(\\iff\\)\u00a0 \\(a_1\\over a_2\\) = \\(b_1\\over b_2\\) = \\(c_1\\over c_2\\)<\/p>\n<\/blockquote>\n
Equation of Straight line making an angle with a Line :<\/strong><\/p>\n
Equation of line passing through a point (\\(x_1,y_1\\)) and making an angle \\(\\alpha\\), with the line y = mx + c is written as<\/p>\n
\ny – \\(y_1\\) = \\(m \\pm tan\\alpha\\over {1 \\mp mtan\\alpha}\\)(x – \\(x_1\\))<\/p>\n<\/blockquote>\n
Reflection and image of a point in a line :<\/h2>\n
Let P(x,y) be any point, then its image with respect to<\/p>\n
(a)\u00a0 x-axis is Q(x,-y)<\/p>\n
(b)\u00a0 y-axis is R(-x, y)<\/p>\n
(c)\u00a0 origin is S(-x,-y)<\/p>\n
(d)\u00a0 line y = x is T(y,x)<\/p>\n
(e)\u00a0 Reflection of a point about any arbitrary line : The image (h,k) of a point P(\\(x_1,y_1\\)) about the line ax+by+c = 0 is given by following formula.<\/p>\n
\n\\(h-x_1\\over a\\) = \\(k-y_1\\over b\\) = -2(\\(ax_1+by_1+c\\over {a^2+b^2}\\))<\/p>\n<\/blockquote>\n
and the foot of perpendicular (p,q) from a point (\\(x_1,y_1\\)) on the line ax+by+c = 0 is given by following formula.<\/p>\n
\n\\(p-x_1\\over a\\) = \\(q-y_1\\over b\\) = -(\\(ax_1+by_1+c\\over {a^2+b^2}\\))<\/p>\n<\/blockquote>\n
Length of perpendicular from a point on a line :<\/h2>\n
Length of perpendicular from a point (\\(x_1,y_1\\)) on the line ax + by + c = 0 is<\/p>\n
\n|\\(ax_1 + by_1 + c\\over {\\sqrt{a^2+b^2}}\\)|<\/p>\n<\/blockquote>\n
In particular, the length of perpendicular from the origin on the line ax + by + c = 0 is<\/p>\n
\nP = \\(|c|\\over {\\sqrt{a^2+b^2}}\\).<\/p>\n<\/blockquote>\n
Hope you learnt formula for angle between two lines and all other concepts. Practice more questions to learn more and get ahead in competition. Good Luck!<\/p>\n\n\n