The curves together; finding a locus

Analytic geometry, grade 12 (5 units) Lesson 19 of 21Learning

The curves together; finding a locus

The equation

\displaystyle {Ax^{2}+By^{2}+Cx+Dy+E}={0}

All the curves of this course, with their axes parallel to the coordinate axes, can be written as $\displaystyle {Ax^{2}+By^{2}+Cx+Dy+E}={0}$ . The coefficients of $\displaystyle x^{2}$ and $\displaystyle y^{2}$ decide which curve it is (if there is one):

$\displaystyle {A}={B}$ – a circle;
$\displaystyle A$ and $\displaystyle B$ of the same sign, different – an ellipse;
$\displaystyle A$ and $\displaystyle B$ of opposite signs – a hyperbola;
one of them $\displaystyle 0$ – a parabola.

Completing the square shows the shift. An example: $\displaystyle {x^{2}+4y^{2}-6x+16y+9}={0}$ .

(x-3)^{2}-9+4(y+2)^{2}-16+9=0\quad\Rightarrow\quad\frac{(x-3)^{2}}{16}+\frac{(y+2)^{2}}{4}=1

An ellipse with centre $\displaystyle (3,-2)$ , $\displaystyle {a}={4}$ and $\displaystyle {b}={2}$ – the canonical ellipse $\displaystyle {\frac{x^{2}}{16}+\frac{y^{2}}{4}}={1}$ , shifted.

There are also degenerate cases: a point, two lines, or no point at all. For example, $\displaystyle {\frac{(x-2)^{2}}{9}}={y^{2}}$ is the two lines $\displaystyle {y}={\pm\frac{x-2}{3}}$ .

\displaystyle x^{2}+4y^{2}-6x+16y+9=0

\displaystyle (3,-2)

Loci that are a parabola, an ellipse or a hyperbola

The locus of the centres of the circles that touch the $\displaystyle x$ -axis and pass through the point $\displaystyle (0,2)$ .

The centre $\displaystyle P(x,y)$ of such a circle is $\displaystyle |y|$ from the $\displaystyle x$ -axis – this is the radius.
The circle passes through $\displaystyle (0,2)$ , so $\displaystyle P$ ’s distance from it is the radius too: $\displaystyle {\sqrt{x^{2}+(y-2)^{2}}}={|y|}$ .
Squared: $\displaystyle {x^{2}+y^{2}-4y+4}={y^{2}}$ , that is $\displaystyle {x^{2}}={4y-4}={4(y-1)}$ .

This is a parabola with vertex $\displaystyle (0,1)$ – as expected: $\displaystyle P$ is as far from the point $\displaystyle (0,2)$ as from the line $\displaystyle {y}={0}$ , which is exactly the definition of a parabola with a focus and a directrix.

In the same way a condition on the sum of the distances from two points gives an ellipse, and a condition on their difference – a hyperbola.

\displaystyle x^{2}=4y-4

\displaystyle (0,2)

Exercise 1

Which curve does each equation describe – a line, two lines, a circle, an ellipse, a hyperbola or a parabola? $\displaystyle {\frac{(x-2)^{2}}{9}}={y^{2}-1}$ , $\displaystyle {\frac{(x-2)^{2}}{9}}={y^{2}+1}$ , $\displaystyle {\frac{(x-2)^{2}}{9}}={y^{2}}$ , $\displaystyle {\frac{(x-2)^{2}}{9}}={-y^{2}+1}$ , $\displaystyle {\frac{(x-2)^{2}}{9}}={y-1}$ .

Exercise 2

Complete the square in

\displaystyle x^{2}-4x

and in

\displaystyle -9y^{2}-18y

Which curve is $\displaystyle {x^{2}-9y^{2}-4x-18y-14}={0}$ ? Find its centre, and complete the square in the card.

/ – 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

The curve and its centre:

Exercise 3

Find the locus of the points as far from $\displaystyle (0,3)$ as from the line $\displaystyle {y}={-3}$ .

\displaystyle y=-3

\displaystyle (0,3)

Exercise 4

Find the locus of the points whose distances from $\displaystyle (0,4)$ and $\displaystyle (0,-4)$ add up to $\displaystyle 10$ .

\displaystyle (0,4)

\displaystyle (0,-4)

Exercise 5

Find the locus of the points whose distance from the point $\displaystyle (4,0)$ is half their distance from the line $\displaystyle {x}={16}$ .

\displaystyle x=16

\displaystyle (4,0)

Which curve is it? What are its vertices?

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