The circle's equation and completing the square

Analytic geometry, grade 12 (5 units) Lesson 6 of 21Learning

The circle's equation and completing the square

The circle as a locus

A circle is the locus of all the points in the plane whose distance from a given point, the centre, is a fixed length, the radius.

A point $\displaystyle P(x,y)$ is on the circle with centre $\displaystyle M(a,b)$ and radius $\displaystyle R$ when $\displaystyle {PM}={R}$ . Squaring gives the general equation of the circle:

\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}

When the centre is the origin, this is grade 10’s canonical circle: $\displaystyle {x^{2}+y^{2}}={R^{2}}$ .
Every circle is a shift of a canonical circle: $\displaystyle a$ to the right and $\displaystyle b$ up, so $\displaystyle x$ is replaced by $\displaystyle x-a$ and $\displaystyle y$ by $\displaystyle y-b$ .

The expanded equation

Opening the brackets of the general equation gives an equation of the form

\displaystyle x^{2}+y^{2}+Cx+Dy+E=0

For example, $\displaystyle {(x-3)^{2}+(y+2)^{2}}={25}$ expands to $\displaystyle {x^{2}+y^{2}-6x+4y-12}={0}$ . In this form the centre and the radius do not show. To get back to the general form we complete the square: $\displaystyle x^{2}-6x$ is the start of the square $\displaystyle {(x-3)^{2}}={x^{2}-6x+9}$ , so $\displaystyle {x^{2}-6x}={(x-3)^{2}-9}$ .

x^{2}-6x+y^{2}+4y-12=(x-3)^{2}-9+(y+2)^{2}-4-12=(x-3)^{2}+(y+2)^{2}-25

The equation is $\displaystyle {(x-3)^{2}+(y+2)^{2}}={25}$ : the centre $\displaystyle (3,-2)$ and the radius $\displaystyle 5$ .

The rule for completing: $\displaystyle {x^{2}+Cx}={\left(x+\frac{C}{2}\right)^{2}-\frac{C^{2}}{4}}$ – half the coefficient of $\displaystyle x$ inside the brackets, and its square taken away.

\displaystyle x^{2}+y^{2}-6x+4y-12=0

\displaystyle M=(3,-2)

When the equation is a circle

An equation of the form $\displaystyle {Ax^{2}+Ay^{2}+Cx+Dy+E}={0}$ – the same coefficient for $\displaystyle x^{2}$ and $\displaystyle y^{2}$ , and no $\displaystyle xy$ term – may be a circle. Divide by $\displaystyle A$ , complete the squares and get $\displaystyle {(x-a)^{2}+(y-b)^{2}}={k}$ :

$\displaystyle k>0$ – a circle of radius $\displaystyle \sqrt{k}$ ;
$\displaystyle {k}={0}$ – a single point, $\displaystyle (a,b)$ : a sum of two squares is $\displaystyle 0$ only when both are $\displaystyle 0$ ;
$\displaystyle k<0$ – no point at all: a sum of squares is never negative.

An example: $\displaystyle {x^{2}+y^{2}-4x+6y+13}={0}$ gives $\displaystyle {(x-2)^{2}+(y+3)^{2}}={0}$ – only the point $\displaystyle (2,-3)$ .

Exercise 1

Write the expression in the form

\displaystyle (x-a)^{2}+(y-b)^{2}-k

The circle: $\displaystyle {x^{2}+y^{2}+8x-2y-8}={0}$ .
Complete the square in the left side.

/ – 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

What are the centre and the radius of the circle?

The radius

/ – 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 divide by

\displaystyle 2

, so that the coefficient of

\displaystyle x^{2}

\displaystyle 1

Find the centre and the radius of the circle $\displaystyle {2x^{2}+2y^{2}-4x+12y+2}={0}$ .
Complete the square in the card, and the centre and the radius go in the box under it.

/ – 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 centre and the radius:

The radius

/ – 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

Which equation is a circle, which a point, and which describes no point at all? $\displaystyle {x^{2}+y^{2}+2x+5}={0}$ , $\displaystyle {x^{2}+y^{2}-10y}={0}$ , $\displaystyle {x^{2}+y^{2}-6x+2y+10}={0}$ .

Exercise 4

Put each point into

\displaystyle {x^{2}+y^{2}+Cx+Dy+E}={0}

. Or: the angle

\displaystyle A

is right, so

\displaystyle BC

is a diameter.

Write the equation of the circle through the points $\displaystyle A(0,0)$ , $\displaystyle B(8,0)$ , $\displaystyle C(0,6)$ .

\displaystyle A=(0,0)

\displaystyle B=(8,0)

\displaystyle C=(0,6)

Exercise 5

A tangent is perpendicular to the radius at the point of contact, so the radius is the centre’s distance from the line.

The centre of the circle is $\displaystyle (2,-1)$ , and the circle touches the line $\displaystyle {3x-4y+5}={0}$ .
Write the equation of the circle.

\displaystyle 3x-4y+5=0

\displaystyle M=(2,-1)

Exercise 6

The centre of the circle is on the $\displaystyle x$ -axis, and the circle passes through $\displaystyle (1,3)$ and $\displaystyle (5,1)$ .

The centre is

\displaystyle (a,0)

, equally far from the two points. Write an equation in

\displaystyle a

and solve it.

Write the equation of the circle.

\displaystyle (1,3)

\displaystyle (5,1)

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