Here you will learn how to find point of intersection of two lines with examples.
Let’s begin –
How to find Point of Intersection of Two Lines
Let the equations of two lines be
\(a_1x + b_1y + c_1\) = 0
and, \(a_2x + b_2y + c_2\) = 0
Suppose these two lines intersect at a point P(\(x_1, y_1\)). Then, (\(x_1, y_1\)) satisfies each of the given equations.
\(\therefore\) \(a_1x_1 + b_1y_1 + c_1\) = 0 and \(a_2x_1 + b_2y_1 + c_2\) = 0
Solving these two by cross-multiplication, we get
\(x_1\over {b_1c_2 – b_2c_1}\) = \(y_1\over {c_1a_2 – c_2a_1}\) = \(1\over {a_1b_2 – a_2b_1}\)
\(\implies\) \(x_1\) = \({b_1c_2 – b_2c_1}\over {a_1b_2 – a_2b_1}\), \(y_1\) = \({c_1a_2 – c_2a_1}\over {a_1b_2 – a_2b_1}\)
Hence the coordinates of the point of the point of intersection of two lines are :
( \({b_1c_2 – b_2c_1}\over {a_1b_2 – a_2b_1}\), \({c_1a_2 – c_2a_1}\over {a_1b_2 – a_2b_1}\))
Formula to find Point of Intersection :
\(x_1\) = ( \({b_1c_2 – b_2c_1}\over {a_1b_2 – a_2b_1}\), \(y_1\) = \({c_1a_2 – c_2a_1}\over {a_1b_2 – a_2b_1}\))
Note : To find the coordinates of the point of intersection of two non-parallel lines, we solve the given equations simultaneously and the values of x and y are so obtained determine the coordinates of the point of intersection.
Example : Find the coordinates of the point of intersecton of the lines 2x – y + 3 = 0 and x + 2y – 4 = 0.
Solution : Solving simultaneously the equations 2x – y + 3 = 0 and x + 2y – 4 = 0, we obtain
\(x\over {4-6}\) = \(y\over {3+8}\) = \(1\over {4+1}\)
\(\implies\) \(x\over -2\) = \(y\over 11\) = \(1\over 5\)
\(\implies\) x = \(-2\over 5\) , y = \(11\over 5\)
Hence, (-2/5, 11/5) is the required point of intersection
Example : Find the coordinates of the point of intersecton of the lines x – y + 4 = 0 and x + 2y – 1 = 0.
Solution : Solving simultaneously the equations x – y + 4 = 0 and x + 2y – 1 = 0, we obtain
\(x\over {1-8}\) = \(y\over {4+1}\) = \(1\over {2+1}\)
\(\implies\) \(x\over -7\) = \(y\over 5\) = \(1\over 3\)
\(\implies\) x = \(-7\over 3\) , y = \(5\over 3\)
Hence, (-7/3, 5/3) is the required point of intersection
Related Questions
Find the coordinates of the point of intersecton of the lines 2x – y + 3 = 0 and x + y – 5 = 0.