Two circles: their positions

Analytic geometry, grade 12 (5 units) Lesson 9 of 21Learning

Two circles: their positions

The distance between the centres

Two circles with radii $\displaystyle R_1\ge R_2$ , and $\displaystyle d$ the distance between their centres. Their position depends on how $\displaystyle d$ compares with the sum of the radii and with their difference:

$\displaystyle d>R_1+R_2$ – the circles are apart, each outside the other; no common point.
$\displaystyle {d}={R_1+R_2}$ – they touch from outside; one common point.
$\displaystyle R_1-R_2<d<R_1+R_2$ – they cross; two common points.
$\displaystyle {d}={R_1-R_2}$ – they touch from inside; one common point.
$\displaystyle d<R_1-R_2$ – the small circle is inside the big one; no common point.

The point of contact is on the line through the two centres. In the drawing: the circle $\displaystyle {(x-5)^{2}+y^{2}}={4}$ touches the circle $\displaystyle {x^{2}+y^{2}}={9}$ from outside, and the dashed circle touches it from inside.

\displaystyle x^{2}+y^{2}=9

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

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

Subtracting the equations: the common chord

Subtracting the two circles’ equations cancels $\displaystyle x^{2}$ and $\displaystyle y^{2}$ and leaves the equation of a line. Every point common to the two circles satisfies it, so when the circles cross, this is the line of the common chord.

An example: $\displaystyle {x^{2}+y^{2}}={25}$ and $\displaystyle {(x-6)^{2}+y^{2}}={13}$ .

$\displaystyle {d}={6}$ , $\displaystyle {R_1}={5}$ , $\displaystyle {R_2}={\sqrt{13}\approx3.6}$ : $\displaystyle 1.4<6<8.6$ , so the circles cross.
Subtract: $\displaystyle {x^{2}+y^{2}-(x^{2}-12x+36+y^{2})}={25-13}$ , that is $\displaystyle {12x-36}={12}$ , and $\displaystyle {x}={4}$ .
Put it into the first equation: $\displaystyle {16+y^{2}}={25}$ , so $\displaystyle {y}={3}$ or $\displaystyle {y}={-3}$ .

The meeting points are $\displaystyle (4,3)$ and $\displaystyle (4,-3)$ , and the common chord is on the line $\displaystyle {x}={4}$ .

\displaystyle x^{2}+y^{2}=25

\displaystyle (x-6)^{2}+y^{2}=13

\displaystyle x=4

Exercise 1

The circles $\displaystyle {x^{2}+y^{2}}={4}$ and $\displaystyle {(x-3)^{2}+(y-4)^{2}}={9}$ .

How do the circles lie? Explain by the distance between the centres.

Find the common points. First subtract the equations.

Exercise 2

The circles $\displaystyle {x^{2}+y^{2}}={25}$ and $\displaystyle {(x-7)^{2}+(y-1)^{2}}={25}$ cross at the points $\displaystyle A$ and $\displaystyle B$ .

The length of the common chord

\displaystyle AB

, in its simplest form

/ – 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 3

You can tell the number of solutions before solving: compare the distance between the centres with the sum of the radii.

How many solutions does the system have? Solve it.

Exercise 4

Mark on the board the points common to the two circles.

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

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

Exercise 5

The system:
$\displaystyle {(x+2)^{2}+(y-1)^{2}}={16}$
$\displaystyle {(x-2)^{2}+(y+2)^{2}}={m^{2}}$ , and $\displaystyle m>0$ .

For which values of $\displaystyle m$ does the system have exactly one solution?

For which values of $\displaystyle m$ does it have two solutions, and for which none?

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