line through an intersection point
Suppose that the lines
have an intersection point. Then for any real value of , the equation
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 and the intersection point of the lines and .
The equation of a line through the common point of those lines is
We have to find such a value for that also lies on the line, i.e. that the equation (3) is satisfied by the values , . So we get for determining the equation
whence . 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 ) and then forming the equation of the line passing this point and the point , but then the calculations would have been substantially longer.
Note. It is apparent that no value of allows the equation (2) to the line
itself. Thus, if we had in the example instead the point e.g. the point of the line , then we had the condition which gives no value of .
- 1 K. Väisälä: Algebran oppi- ja esimerkkikirja II. Neljäs painos. Werner Söderström osakeyhtiö, Porvoo & Helsinki (1956).
|Title||line through an intersection point|
|Date of creation||2013-03-22 17:30:28|
|Last modified on||2013-03-22 17:30:28|
|Last modified by||pahio (2872)|