The distance of a point from a line; the positions of two lines

Analytic geometry, grade 12 (5 units) Lesson 3 of 21Learning

The distance of a point from a line; the positions of two lines

The distance between two objects

The distance between two geometric objects is the shortest distance between a point of one and a point of the other.

The distance of a point $\displaystyle P$ from a line is the length of the perpendicular $\displaystyle PH$ from $\displaystyle P$ to the line: any other segment $\displaystyle PQ$ to the line is the hypotenuse of the right-angled triangle $\displaystyle PHQ$ , so it is longer than $\displaystyle PH$ .
The distance between two parallel lines is the distance of any point on one from the other – it is the same for every point.

In grade 10 we found $\displaystyle PH$ in three steps: the perpendicular’s equation, the meeting point $\displaystyle H$ , and the distance $\displaystyle PH$ . One formula does all three together.

The formula for a point’s distance from a line

The formula needs the line in implicit form.

\displaystyle {y}={2x}

is written

\displaystyle {2x-y}={0}

\displaystyle {A}={2}

\displaystyle {B}={-1}

\displaystyle {C}={0}

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}}}

In the numerator the point is put into the left side of the line’s equation. A point on the line gives $\displaystyle 0$ – a distance of $\displaystyle 0$ .
The absolute value – because a distance is never negative.

An example: the point $\displaystyle K(3,1)$ and the line $\displaystyle {y}={2x}$ , that is $\displaystyle {2x-y}={0}$ :

d=\frac{|2\cdot3-1\cdot1+0|}{\sqrt{2^{2}+(-1)^{2}}}=\frac{5}{\sqrt{5}}=\sqrt{5}

This is the distance grade 10 found in three steps: the perpendicular meets the line at $\displaystyle H(1,2)$ , and $\displaystyle {KH}={\sqrt{5}}$ .

\displaystyle 2x-y=0

\displaystyle y=-\frac{1}{2}x+\frac{5}{2}

\displaystyle K=(3,1)

\displaystyle H=(1,2)

The distance between parallel lines

When the lines are written with the same $\displaystyle A$ and $\displaystyle B$ , $\displaystyle {Ax+By+C_1}={0}$ and $\displaystyle {Ax+By+C_2}={0}$ , the distance between them is:

\displaystyle d

\displaystyle =

\displaystyle \frac{|C_1-C_2|}{\sqrt{A^{2}+B^{2}}}

An example: $\displaystyle {3x-4y+5}={0}$ and $\displaystyle {6x-8y-15}={0}$ .

We divide the second equation by $\displaystyle 2$ to get the same $\displaystyle A$ and $\displaystyle B$ : $\displaystyle {3x-4y-7.5}={0}$ .
$\displaystyle {d}={\frac{|5-(-7.5)|}{\sqrt{3^{2}+4^{2}}}}={\frac{12.5}{5}}={2.5}$ .

The positions of two lines

Two lines in the plane have three positions, and the number of solutions of the system of their equations tells them apart:

crossing – different slopes; the system has one solution, the meeting point.
parallel – the same slope, different lines; the system has no solution (an equality like $\displaystyle {0}={7}$ appears).
coinciding – the same line in two equations, such as $\displaystyle {x+2y}={4}$ and $\displaystyle {2x+4y}={8}$ ; the system has infinitely many solutions ( $\displaystyle {0}={0}$ appears).

Exercise 1

Work out the distance of the point $\displaystyle P(2,-1)$ from the line $\displaystyle {3x+4y-12}={0}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

First write the line in implicit form. Write the result with no root in the denominator.

Work out the distance of the point $\displaystyle P(1,5)$ from the line $\displaystyle {y}={2x-3}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Work out the distance between the parallel lines $\displaystyle {x-2y+3}={0}$ and $\displaystyle {2x-4y-14}={0}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

Solve each system, and say how the two lines lie.

How each pair of lines lies:

Exercise 5

Write an equation in

\displaystyle k

from the distance formula. An absolute value gives two possibilities.

The distance of the point $\displaystyle (k,1)$ from the line $\displaystyle {4x+3y-7}={0}$ is $\displaystyle 2$ .
Find $\displaystyle k$ .

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
$\displaystyle y=\pm\frac{b}{a}x$