If three lines whose equations are concurrent, then show that
The equation of given lines is,
y = m 1 x + c 1 (1)
y = m 2 x + c 2 (2)
y = m 3 x + c 3 (3)
Subtract equation (1) from equation (2).
0 = ( m 2 − m 1 ) x + ( c 2 − c 1 ) ( m 2 − m 1 ) x = − ( c 2 − c 1 ) x = − ( c 2 − c 1 ) ( m 2 − m 1 ) x = ( c 2 − c 1 ) ( m 1 − m 2 )
Substitute the value of x in equation (1).
y = m 1 × ( c 2 − c 1 ) ( m 1 − m 2 ) + c 1 = m 1 c 2 − m 1 c 1 m 1 − m 2 + c 1 = m 1 c 2 − m 1 c 1 + c 1 × ( m 1 − m 2 ) m 1 − m 2 = m 1 c 2 − m 1 c 1 + c 1 m 1 − c 1 m 2 m 1 − m 2
Further solve the above expression.
y = m 1 c 2 − m 2 c 1 m 1 − m 2
The coordinate of the point of intersection of line (1) and line (2) is ( c 2 − c 1 m 1 − m 2 , m 1 c 2 − m 2 c 1 m 1 − m 2 ) .
It is given in the question that line (1), line (2) and line (3) are concurrent.
Thus, the point ( c 2 − c 1 m 1 − m 2 , m 1 c 2 − m 2 c 1 m 1 − m 2 ) will also satisfy the equation of line (3).
Substitute the value of point in equation of line (3).
m 1 c 2 − m 2 c 1 m 1 − m 2 = m 3 c 2 − c 1 m 1 − m 2 + c 3 m 1 c 2 − m 2 c 1 m 1 − m 2 = m 3 c 2 − m 3 c 1 + c 3 ( m 1 − m 2 ) m 1 − m 2 m 1 c 2 − m 2 c 1 = m 3 c 2 − m 3 c 1 + c 3 m 1 − c 3 m 2 m 1 c 2 − m 2 c 1 − m 3 c 2 + m 3 c 1 − c 3 m 1 + c 3 m 2 = 0
Further simplify the above expression.
m 1 ( c 2 − c 3 ) + m 2 ( c 3 − c 1 ) + m 3 ( c 1 − c 2 ) = 0
Hence, the condition for the three lines y = m 1 x + c 1 , y = m 2 x + c 2 , and y = m 3 x + c 3 to be concurrent is m 1 ( c 2 − c 3 ) + m 2 ( c 3 − c 1 ) + m 3 ( c 1 − c 2 ) = 0 .
The equation of given lines is,
Subtract equation (1) from equation (2).
Substitute the value of
Further solve the above expression.
The coordinate of the point of intersection of line (1) and line (2) is
It is given in the question that line (1), line (2) and line (3) are concurrent.
Thus, the point
Substitute the value of point in equation of line (3).
Further simplify the above expression.
Hence, the condition for the three lines
