Families of Lines

What this move does

A line needs two conditions to be determined. If only one condition is given, infinitely many lines satisfy it, and together they form a family of lines:

lines through the intersection of l_{1}, l_{2} (without l_{2}) (A_{1} x + B_{1} y + C_{1}) + λ (A_{2} x + B_{2} y + C_{2}) = 0 lines parallel to A x + B y + C = 0 A x + B y + C^{'} = 0

Why do all the lines in the first family pass through the intersection point? Substituting the intersection point $P_{0}$ into the left sides of $l_{1}$ and $l_{2}$ gives $0$ for both, so $0 + λ \cdot 0 = 0$ holds for every $λ$ . Solving a problem takes three steps:

From “through the intersection of $l_{1}$ and $l_{2}$ ” (or “parallel to a given line”), write down the family. There is no need to find the intersection point.
Use the second condition (through a given point, perpendicular to a given line…) to get an equation in $λ$ .
Solve for $λ$ , substitute back and simplify. If $λ$ has no solution, check whether $l_{2}$ itself satisfies the condition.

Predict first

Whatever real value $m$ takes, the line $(1 + 2 m) x + (m - 1) y + m - 7 = 0$ passes through one fixed point. Which point?

(2, - 5)

(- 5, 2)

(1, 1)

There is no such point

Choose an answer first; then you can see the result.

Work through one

Example: the line through the intersection of x − y + 1 = 0 and x + y − 3 = 0 and through T(3, 5)

Step 1 / 5

Write down the family
With $λ = 0$ we get $l_{1}$ itself. The thin gray lines are the members for other values of $λ$ ; they all pass through the same point.

$(x - y + 1) + λ (x + y - 3) = 0$
They all pass through the center
Use the second condition to find λ
Substitute back and simplify
When λ is large

A family of parallel lines is even simpler: only the constant term is a parameter.

The family of lines parallel to

3 x - 2 y = 0

. Drag

T

, then click “Pass through T”:

C^{'} = - (3 x_{T} - 2 y_{T})

When it fails

Break it

Counterexample lab: remove one condition

Conclusion: Every member of $l_{1} + λ l_{2} = 0$ is a line through one common point

When they intersect, every value of $λ$ gives a line through the center (it just never gives $l_{2}$ ).

Practice

Exercise 1Basic

Whatever value $k$ takes, the line $k x - y + 2 k + 3 = 0$ passes through a fixed point. Find it. (In the form (a,b).)

Exercise 2Basic

Find the equation of the line through the intersection of $2 x + y - 5 = 0$ and $x - 3 y + 8 = 0$ that also passes through the origin (general form, positive coefficient of $x$ , coefficients with no common factor).

Exercise 3Advanced

Find the equation of the line through the intersection of $x + y - 2 = 0$ and $2 x - y - 1 = 0$ that is perpendicular to $x + 3 y + 5 = 0$ (general form, positive coefficient of $x$ , coefficients with no common factor).

Exercise 4Challenge

In the family $(1 + λ) x + (2 - λ) y + λ - 5 = 0$ , find the values of $λ$ for which the line has equal intercepts on the two axes (separate them with commas).

What this move does

Work through one

Example: the line through the intersection of x − y + 1 = 0 and x + y − 3 = 0 and through T(3, 5)

When it fails

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Setting up l₁ + λl₂ = 0 and losing l₂