# line through an intersection point

Suppose that the lines

 $\displaystyle Ax+By+C=0\;\;\mbox{and}\;\;A^{\prime}x+B^{\prime}y+C^{\prime}=0$ (1)

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

 $\displaystyle Ax+By+C+k(A^{\prime}x+B^{\prime}y+C^{\prime})=0$ (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

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

 $-35-10k=0,$

whence  $k=-\frac{7}{2}$. Using this value in (3), multiplying the equation by 2 and simplifying, we obtain the sought equation

 $23x+77y-39=0.$

This result would be obtained, of course, by first calculating the intersection point of the two given lines (it is  $(-\frac{60}{89},\,\frac{63}{89})$) 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
$A^{\prime}x+B^{\prime}y+C^{\prime}=0$  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 $k$.

## References

• 1 K. Väisälä: Algebran oppi- ja esimerkkikirja II. Neljäs painos.   Werner Söderström osakeyhtiö, Porvoo & Helsinki (1956).
Title line through an intersection point LineThroughAnIntersectionPoint 2013-03-22 17:30:28 2013-03-22 17:30:28 pahio (2872) pahio (2872) 8 pahio (2872) Topic msc 53A04 msc 51N20