# line through an intersection point

Suppose that the lines

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

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

$Ax+By+C+k({A}^{\prime}x+{B}^{\prime}y+{C}^{\prime})=0$ | (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 $(-5,\mathrm{\hspace{0.17em}2})$ 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 $(-5,\mathrm{\hspace{0.17em}2})$ 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 $(-5,\mathrm{\hspace{0.17em}2})$, 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 $(-5,\mathrm{\hspace{0.17em}2})$ 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 |
---|---|

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 |