Loci that are lines

Analytic geometry, grade 12 (5 units) Lesson 4 of 21Learning

Loci that are lines

A locus

A locus is the set of all the points in the plane that satisfy a condition. In analytic geometry it is found in three steps:

Any point of the locus is $\displaystyle P(x,y)$ .
Write the condition as an equation in $\displaystyle x$ and $\displaystyle y$ .
Simplify the equation until the shape can be recognised: a line, a circle, …

Every step of the simplifying is an equivalence, so the equation reached describes the locus exactly: every point that satisfies the condition is on the shape, and every point on the shape satisfies the condition. In grade 10’s geometry we proved these two directions separately.

An example: the perpendicular bisector

The locus of all the points whose distance from $\displaystyle A(1,2)$ equals their distance from $\displaystyle B(5,4)$ .

The condition is $\displaystyle {PA}={PB}$ . We square, to get rid of the roots:

(x-1)^{2}+(y-2)^{2}=(x-5)^{2}+(y-4)^{2}

x^{2}-2x+1+y^{2}-4y+4=x^{2}-10x+25+y^{2}-8y+16

$\displaystyle x^{2}$ and $\displaystyle y^{2}$ cancel, leaving $\displaystyle {8x+4y}={36}$ , that is $\displaystyle {2x+y}={9}$ .

This is a line: the perpendicular bisector of $\displaystyle AB$ .
A check: the midpoint $\displaystyle (3,3)$ is on it, and its slope $\displaystyle -2$ is perpendicular to $\displaystyle AB$ ’s slope $\displaystyle \frac{1}{2}$ .

\displaystyle 2x+y=9

\displaystyle y=\frac{1}{2}x+\frac{3}{2}

\displaystyle A=(1,2)

\displaystyle B=(5,4)

An example: a given distance from a line

The locus of all the points whose distance from the line $\displaystyle {3x+4y-1}={0}$ is $\displaystyle 2$ .

\frac{|3x+4y-1|}{5}=2\quad\Rightarrow\quad|3x+4y-1|=10

An absolute value gives two possibilities: $\displaystyle {3x+4y-1}={10}$ or $\displaystyle {3x+4y-1}={-10}$ . The locus is two lines parallel to the given one, one on each side of it: $\displaystyle {3x+4y}={11}$ and $\displaystyle {3x+4y}={-9}$ .

\displaystyle 3x+4y-1=0

\displaystyle 3x+4y=11

\displaystyle 3x+4y=-9

An example: the bisectors of the angles between two lines

In grade 10 we proved: an angle bisector is the locus of the points inside the angle that are equally far from its two arms. So the locus of the points equally far from two crossing lines is the two bisectors of the angles between them.

The lines $\displaystyle {3x-4y}={0}$ and $\displaystyle {4x-3y}={0}$ :

\frac{|3x-4y|}{5}=\frac{|4x-3y|}{5}\quad\Rightarrow\quad3x-4y=\pm(4x-3y)

with $\displaystyle +$ : $\displaystyle {-x-y}={0}$ , that is $\displaystyle {y}={-x}$ ;
with $\displaystyle -$ : $\displaystyle {7x-7y}={0}$ , that is $\displaystyle {y}={x}$ .

The two bisectors are perpendicular to each other – as always between crossing lines.

\displaystyle 3x-4y=0

\displaystyle 4x-3y=0

\displaystyle y=x

\displaystyle y=-x

Exercise 1

$\displaystyle A(-2,1)$ and $\displaystyle B(4,-3)$ .
Find the locus of all the points whose distance from $\displaystyle A$ equals their distance from $\displaystyle B$ , and write its equation in a row of your own on the board.

\displaystyle A=(-2,1)

\displaystyle B=(4,-3)

Exercise 2

Find the locus of all the points whose distance from the line $\displaystyle {4x-3y+2}={0}$ is $\displaystyle 3$ .
Write its equations on the board, each line in a row of its own.

\displaystyle 4x-3y+2=0

Exercise 3

The area gives the height to the side

\displaystyle AB

, and the height is the distance of the third vertex from the line

\displaystyle AB

$\displaystyle A(2,0)$ and $\displaystyle B(1,3)$ are two vertices of a triangle $\displaystyle ABC$ .
The triangle’s area is $\displaystyle 4$ .
What is the locus of the vertex $\displaystyle C$ ? Write its equations on the board.

\displaystyle 3x+y-6=0

\displaystyle A=(2,0)

\displaystyle B=(1,3)

Exercise 4

Find the bisectors of the angles between the $\displaystyle x$ -axis and the line $\displaystyle {4x-3y}={0}$ .
Write their equations on the board.

\displaystyle 4x-3y=0

Formulas: analytic geometry, grade 12 (5 units)

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Points and lines

The distance between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}}$
The slope of the line through the two points
$\displaystyle m$ $\displaystyle =$ $\displaystyle \frac{y_2-y_1}{x_2-x_1}$ $\displaystyle x_1\neq x_2$
The line with slope $\displaystyle m$ through $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1=m(x-x_1)$
Perpendicular lines – $\displaystyle m_1$ , $\displaystyle m_2$ their slopes
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$
The point $\displaystyle C$ dividing the segment $\displaystyle AB$ in the ratio $\displaystyle {AC:CB}={k:l}$ – $\displaystyle A(x_1,y_1)$ , $\displaystyle B(x_2,y_2)$
$\displaystyle C\left(\frac{lx_1+kx_2}{k+l},\frac{ly_1+ky_2}{k+l}\right)$
The distance of the point $\displaystyle (x_0,y_0)$ from the line $\displaystyle {Ax+By+C}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|Ax_0+By_0+C|}{\sqrt{A^{2}+B^{2}}}$
The distance between the parallel lines $\displaystyle {Ax+By+C_1}={0}$ and $\displaystyle {Ax+By+C_2}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|C_1-C_2|}{\sqrt{A^{2}+B^{2}}}$

Circle

A circle with centre $\displaystyle (a,b)$ and radius $\displaystyle R$
$\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}$
The expanded equation – a circle when, after completing the square, the right side is positive
$\displaystyle x^{2}+y^{2}+Cx+Dy+E=0$
The tangent to the circle at the point $\displaystyle (x_0,y_0)$ on it
$\displaystyle (x_0-a)(x-a)+(y_0-b)(y-b)=R^{2}$

Parabola

The canonical parabola
$\displaystyle y^{2}=2px$
The focus
$\displaystyle F\left(\frac{p}{2},0\right)$
The directrix
$\displaystyle x=-\frac{p}{2}$
The tangent at the point $\displaystyle (x_0,y_0)$ on the parabola
$\displaystyle y\cdot y_0=p(x+x_0)$

Ellipse

The canonical ellipse – the foci on the $\displaystyle x$ -axis when $\displaystyle a>b$
$\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
The distance of a focus from the origin
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^{2}-b^{2}}$
The sum of the distances of a point $\displaystyle P$ on the ellipse from the foci
$\displaystyle PF_1+PF_2$ $\displaystyle =$ $\displaystyle 2a$

Hyperbola

The canonical hyperbola – the foci on the $\displaystyle x$ -axis
$\displaystyle \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
The distance of a focus from the origin
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^{2}+b^{2}}$
The difference of the distances of a point $\displaystyle P$ on the hyperbola from the foci
$\displaystyle |PF_1-PF_2|$ $\displaystyle =$ $\displaystyle 2a$
The asymptotes
$\displaystyle y=\pm\frac{b}{a}x$