line through an intersection point

Suppose that the lines

Ax+By+C=0andAx+By+C=0 (1)

have an intersection point. Then for any real value of k, the equation

Ax+By+C+k(Ax+By+C)=0 (2)

represents a line passing through that point.

In fact, the of the equation (2) is 1, and therefore it represents a line; secondly, (2) is satisfied if both equations (1) are satisfied, and therefore the line passes through that intersection point.

Example. Determine the equation of the line passing through the point  (-5, 2)  and the intersection point of the lines  6x-7y+9=0  and  5x+9y-3=0.

The equation of a line through the common point of those lines is

6x-7y+9+k(5x+9y-3)=0. (3)

We have to find such a value for k that also  (-5, 2)  lies on the line, i.e. that the equation (3) is satisfied by the values  x=-5,  y=2. So we get for determining k the equation


whence  k=-72. Using this value in (3), multiplying the equation by 2 and simplifying, we obtain the sought equation


This result would be obtained, of course, by first calculating the intersection point of the two given lines (it is  (-6089,6389)) and then forming the equation of the line passing this point and the point  (-5, 2), but then the calculations would have been substantially longer.

Note. It is apparent that no value of k allows the equation (2) to the line 
Ax+By+C=0  itself. Thus, if we had in the example instead the point  (-5, 2)  e.g. the point  (6,-3)  of the line  5x+9y-3=0, then we had the condition  66+0k=0  which gives no value of k.


  • 1 K. Väisälä: Algebran oppi- ja esimerkkikirja II. Neljäs painos.   Werner Söderström osakeyhtiö, Porvoo & Helsinki (1956).
Title line through an intersection point
Canonical name LineThroughAnIntersectionPoint
Date of creation 2013-03-22 17:30:28
Last modified on 2013-03-22 17:30:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Topic
Classification msc 53A04
Classification msc 51N20