Family of Lines

5.4.4 Family of Lines

A collection of lines sharing one common property is called a family of lines. For instance, the set of all lines through the origin is a family — every member has equation $y=mx$ for some value of $m$, and different values of $m$ pick out different lines. Likewise, the set of all lines through a fixed point $A(2,-3)$ is a family, each of the form $y+3=m(x-2)$; and the set of all lines parallel to $y=x$ (all sharing slope $1$) is a family too. In general: the set of all lines through a fixed point, or all mutually parallel to each other, is a family of lines.

Interpreting $u+kv=0$. Let $u \equiv a_1x+b_1y+c_1$ and $v\equiv a_2x+b_2y+c_2$, so $u=0$ and $v=0$ are two given lines. Then, for any real $k$, the equation

represents a family of lines. Substituting, $a_1x+b_1y+c_1+k(a_2x+b_2y+c_2)=0$ rearranges to $(a_1+ka_2)x+(b_1+kb_2)y+(c_1+kc_2)=0$, which is a first-degree equation in $x,y$ for every fixed $k$, so it always represents some straight line.

(i) If $u=0$ and $v=0$ intersect at $P(x_1,y_1)$, then $a_1x_1+b_1y_1+c_1=0$ and $a_2x_1+b_2y_1+c_2=0$ simultaneously. Substituting $P$ into $u+kv$: $(a_1x_1+b_1y_1+c_1)+k(a_2x_1+b_2y_1+c_2)=0+k(0)=0$. So every member of the family $u+kv=0$ passes through the same fixed point $P$, the intersection of the two original lines — regardless of the value of $k$.

(ii) If $u=0$ and $v=0$ are parallel, their slopes agree: $-\dfrac{a_1}{b_1}=-\dfrac{a_2}{b_2}$. Then the slope of $u+kv=0$, namely $-\dfrac{a_1+ka_2}{b_1+kb_2}$, works out (using that common ratio) to equal the same slope as $u=0$ and $v=0$. So the whole family $u+kv=0$ is parallel to both original lines. …