Complete study of a pencil

Problem

In the pencil $k(x+2y-1) + (x-y+2) = 0$:

• (a) determine the base point;
• (b) find the line of the pencil parallel to the $x$-axis;
• (c) find the line parallel to $y = 3x-5$;
• (d) find the line at distance $2$ from the origin.

Solution

The two generators are $r_1: x+2y-1 = 0$ (the one multiplying $k$) and $r_2: x-y+2 = 0$ (the part independent of $k$). Rearranging the equation of the pencil in $x$, $y$: $(1+k)\,x + (2k-1)\,y + (2-k) = 0.$

(a) Base point. It is the point of intersection of the generators. From $x-y+2=0$ we get $x=y-2$; substituting into $x+2y-1=0$: $(y-2)+2y-1=0 \implies 3y-3=0 \implies y=1,\quad x=-1.$ The generators are incident (slopes $-\tfrac12$ and $1$), so the pencil is proper with base point $\boxed{P(-1;1)}$.

(b) Parallel to the $x$-axis. A horizontal line has a zero coefficient of $x$: $1+k=0 \implies k=-1$. Substituting: $0\cdot x + (2(-1)-1)\,y + (2-(-1)) = 0 \implies -3y+3=0 \implies \boxed{y=1}.$ (The excluded generator $r_1$ is not horizontal, so it adds no solutions.)

(c) Parallel to $y=3x-5$. We need slope $3$. The slope of the line of the pencil is $-\dfrac{1+k}{2k-1}$, so: $-\frac{1+k}{2k-1}=3 \implies -(1+k)=3(2k-1) \implies -1-k=6k-3 \implies 7k=2 \implies k=\tfrac{2}{7}.$ Substituting and multiplying by $7$: $9x-3y+12=0$, that is $\boxed{y=3x+4}$.

(d) Distance $2$ from the origin. The distance of $O(0;0)$ from the line of the pencil is $d=\frac{|2-k|}{\sqrt{(1+k)^2+(2k-1)^2}}=\frac{|2-k|}{\sqrt{5k^2-2k+2}}.$ Imposing $d=2$ and squaring: $(2-k)^2 = 4(5k^2-2k+2)$, that is $19k^2-4k+4=0, \qquad \Delta = 16-304 = -288 < 0.$ No real value of $k$. We also check the excluded generator $r_1: x+2y-1=0$, whose distance from $O$ is $\dfrac{1}{\sqrt5}\approx 0{,}45 \ne 2$. The reason is geometric: every line of the pencil passes through $P(-1;1)$, and the distance of $O$ from a line through $P$ cannot exceed $|OP|=\sqrt2\approx 1{,}41 < 2$. Hence $\boxed{\text{no line of the pencil is at distance }2\text{ from the origin}}$.

Links

Topics: Pencils of lines
Concepts: Point–line distance · Pencil of lines · Proper pencil · Generators · Parallelism · Pencil parameter · Line · Base point of the pencil
Functions: Line
Skills: Analytic geometry · Reasoning by cases · Solving equations · Using formulae
Exercise type: Geometric problem