## You are here

Homeline through an intersection point

## Primary tabs

# 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 degree 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 represent 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).

## Mathematics Subject Classification

53A04*no label found*51N20

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections