The parabola: a locus and its canonical equation

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

The parabola: a locus and its canonical equation

Focus and directrix

A parabola is the locus of all the points in the plane whose distance from a given point, the focus $\displaystyle F$ , equals their distance from a given line, the directrix.

To get a simple equation, we choose the coordinate system like this:

the $\displaystyle x$ -axis passes through the focus, perpendicular to the directrix;
the origin is halfway between the focus and the directrix.

If the distance between the focus and the directrix is $\displaystyle p$ , the focus is $\displaystyle F\left(\frac{p}{2},0\right)$ and the directrix $\displaystyle {x}={-\frac{p}{2}}$ . The distance of $\displaystyle P(x,y)$ from the directrix is $\displaystyle \left|x+\frac{p}{2}\right|$ , so the condition is:

\sqrt{\left(x-\frac{p}{2}\right)^{2}+y^{2}}=\left|x+\frac{p}{2}\right|

Squaring: $\displaystyle {x^{2}-px+\frac{p^{2}}{4}+y^{2}}={x^{2}+px+\frac{p^{2}}{4}}$ , and almost everything cancels. What is left is the canonical equation of the parabola:

\displaystyle y^{2}=2px

Properties from the equation

From the equation $\displaystyle {y^{2}}={2px}$ we see:

Symmetry – if $\displaystyle (x,y)$ is on the parabola, so is $\displaystyle (x,-y)$ : $\displaystyle y$ appears only squared. The $\displaystyle x$ -axis is the axis of symmetry.
The vertex – the origin, the parabola’s point nearest the directrix.
The direction – when $\displaystyle p>0$ , $\displaystyle {x}={\frac{y^{2}}{2p}\ge0}$ : the parabola opens to the right, towards the focus. When $\displaystyle p<0$ it opens to the left.

An example: $\displaystyle {y^{2}}={12x}$ . Here $\displaystyle {2p}={12}$ , so $\displaystyle {p}={6}$ : the focus is $\displaystyle F(3,0)$ and the directrix $\displaystyle {x}={-3}$ . The point $\displaystyle (3,6)$ is on the parabola, $\displaystyle 6$ away both from the focus and from the directrix.

\displaystyle y^{2}=12x

\displaystyle x=-3

\displaystyle F=(3,0)

\displaystyle (3,6)

A focus on the

\displaystyle y

-axis

When the $\displaystyle y$ -axis is chosen through the focus, $\displaystyle x$ and $\displaystyle y$ swap: the focus is $\displaystyle F\left(0,\frac{p}{2}\right)$ , the directrix $\displaystyle {y}={-\frac{p}{2}}$ , and the equation $\displaystyle {x^{2}}={2py}$ .

This is grade 9’s parabola: $\displaystyle {y}={\frac{1}{2p}x^{2}}$ . For example, $\displaystyle {y}={\frac{1}{4}x^{2}}$ , that is $\displaystyle {x^{2}}={4y}$ , has $\displaystyle {p}={2}$ : the focus $\displaystyle (0,1)$ and the directrix $\displaystyle {y}={-1}$ .

A shifted parabola

When the focus and the directrix are not in the canonical place, the same method gives a shifted parabola.

An example: the focus $\displaystyle (3,2)$ and the directrix $\displaystyle {x}={-1}$ .

(x-3)^{2}+(y-2)^{2}=(x+1)^{2}\quad\Rightarrow\quad(y-2)^{2}=8x-8=8(x-1)

This is the parabola $\displaystyle {y^{2}}={8x}$ shifted $\displaystyle 1$ to the right and $\displaystyle 2$ up: the vertex is $\displaystyle (1,2)$ , halfway between the focus and the directrix.

Exercise 1

The parabola: $\displaystyle {y^{2}}={8x}$ .

What are $\displaystyle p$ and the focus?

\displaystyle p

/ – 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

Write the directrix’s equation on the board.

\displaystyle y^{2}=8x

Exercise 2

A parabola’s focus is $\displaystyle F(5,0)$ , and its directrix $\displaystyle {x}={-5}$ .
Write the parabola’s equation.

\displaystyle x=-5

\displaystyle F=(5,0)

Exercise 3

A parabola’s focus is $\displaystyle (0,2)$ , and its directrix $\displaystyle {y}={-2}$ .
Write the parabola’s equation.

\displaystyle y=-2

\displaystyle F=(0,2)

Exercise 4

By the definition, the distance from the focus equals the distance from the directrix – and that one is easy to work out.

The point $\displaystyle P$ is on the parabola $\displaystyle {y^{2}}={16x}$ , and $\displaystyle {x_P}={9}$ .

What is $\displaystyle y_P$ ? (There are two such points.)

The distance of

\displaystyle P

from the focus

/ – 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 5

A parabola’s focus is $\displaystyle (4,-1)$ , and its directrix $\displaystyle {x}={0}$ .
Write the parabola’s equation.

\displaystyle x=0

\displaystyle F=(4,-1)

Exercise 6

The parabola $\displaystyle {y^{2}}={2px}$ passes through the point $\displaystyle (2,-4)$ .

Find

\displaystyle p

What are the focus and the directrix of the parabola?

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