Circles and Ellipses

Lesson

We have looked at the general equation of a circle, $\left(x-h\right)^2+\left(y-k\right)^2=r^2$(`x`−`h`)2+(`y`−`k`)2=`r`2, which is centred at the point $\left(h,k\right)$(`h`,`k`) and has a radius of $r$`r`. A special case of this equation occurs when the circle is centred at the origin. In this case, $h$`h` and $k$`k` are both $0$0 and so the equation takes the form $x^2+y^2=r^2$`x`2+`y`2=`r`2.

We can make use of our knowledge of circles and their equations to solve various problems involving circles. Let's take a look at a few properties.

- The equation of a circle describes all of the points that lie
**exactly**a distance of $r$`r`away from the centre point $\left(h,k\right)$(`h`,`k`). Any points which are closer than $r$`r`units to the centre lie**inside**the circle, and any points that are further than $r$`r`units from the centre lie**outside**the circle.

- Two points which lie on opposite ends of a diameter of a circle have the centre of the circle as their midpoint.

- The area of a circle of radius $r$
`r`is given by $A=\pi r^2$`A`=π`r`2. Similarly, the length of the circumference of a circle of radius $r$`r`is given by $C=2\pi r$`C`=2π`r`. - A circle can be inscribed in a square as in the image below. This square is the smallest possible square that can fit around the circle, which touches at four points and has side lengths equal to the length of the diameter of the circle.

Let's go through an example now.

A circle of radius $3$3 units is centred at the point $\left(12,5\right)$(12,5). What is the shortest distance from the origin to the circle?

Think: We can start by drawing a diagram of the situation, to easily see what's going on.

From the diagram, we can see that the closest point to the origin is between the centre and the origin. In fact, if we draw a line from the origin to the centre of the circle it will intersect the circle at the closest point.

This means that the shortest distance from the origin to the circle will be equal to the distance between the origin and the centre, minus the radius of the circle.

Do: We know that the radius of the circle is $3$3 units, and we can use the distance formula to find the distance between the origin and the centre:

$d$d |
$=$= | $\sqrt{\left(12-0\right)^2+\left(5-0\right)^2}$√(12−0)2+(5−0)2 |

$=$= | $\sqrt{12^2+5^2}$√122+52 | |

$=$= | $\sqrt{144+25}$√144+25 | |

$=$= | $\sqrt{169}$√169 | |

$=$= | $13$13 |

Subtracting the radius from this, we can see that the shortest distance from the origin to the circle is $13-3=10$13−3=10 units.

Consider a circle with centre $O$`O` at $\left(0,0\right)$(0,0) that passes through the point $A\left(-7,6\right)$`A`(−7,6). A diameter is drawn from point $A$`A` to point $B$`B` on the circle.

State the coordinates of point $B$`B`.

Naval ship $A$`A` picks up the signal of another vessel $45$45 km away. Ship $A$`A` can also see that there is a partner naval ship, ship $B$`B`, $30$30 km south and $40$40 km west of it.

Using $\left(0,0\right)$(0,0) as the position of naval ship $A$

`A`, determine the distance between naval ships $A$`A`and $B$`B`.Could the signal that was picked up be coming from naval ship $B$

`B`?No

AYes

BNo

AYes

B

Consider the circle that has been graphed.

Find the area of the square that inscribes the circle (contains the circle exactly).

Loading Graph...

We know that the general equation of a circle is $\left(x-h\right)^2+\left(y-k\right)^2=r^2$(`x`−`h`)2+(`y`−`k`)2=`r`2, which is centred at the point $\left(h,k\right)$(`h`,`k`) and has a radius of $r$`r`.

From this we can find the equations of the upper- and lower-half semicircles by solving for $y$`y`.

Upper: $y=k+\sqrt{r^2-\left(x-h\right)^2}$`y`=`k`+√`r`2−(`x`−`h`)2

Lower: $y=k-\sqrt{r^2-\left(x-h\right)^2}$`y`=`k`−√`r`2−(`x`−`h`)2

Similarly, the equations of the left- and right-half semicircles can be found by solving for $x$`x`.

Right: $x=h+\sqrt{r^2-\left(y-k\right)^2}$`x`=`h`+√`r`2−(`y`−`k`)2

Left: $x=h-\sqrt{r^2-\left(y-k\right)^2}$`x`=`h`−√`r`2−(`y`−`k`)2

In the special case that the semicircles are centred at the origin, so $h=0$`h`=0 and $k=0$`k`=0, these equations simplify to $y=\pm\sqrt{r^2-x^2}$`y`=±√`r`2−`x`2 and $x=\pm\sqrt{r^2-y^2}$`x`=±√`r`2−`y`2.

We can make use of these equations, and our knowledge of circles, to solve various problems involving semicircles. Let's take a look at a few properties.

- The equation of a semicircle describes a set of points that lie
**exactly**a distance of $r$`r`away from the centre point $\left(h,k\right)$(`h`,`k`), and are to one side of the centre. Any points on the same side which are closer than $r$`r`units to the centre lie**inside**the semicircle, and any points that are further than $r$`r`units from the centre**or**on the other side lie**outside**the semicircle.

- Any diameter divides a circle into two semicircles. The two endpoints of the diameter have the centre of the circle as their midpoint.

- The area of a circle of radius $r$
`r`is given by $A=\pi r^2$`A`=π`r`2. Similarly, the length of the circumference of a circle of radius $r$`r`is given by $C=2\pi r$`C`=2π`r`.

Correspondingly, the area of a semicircle of radius $r$`r`is given by $A=\frac{1}{2}\pi r^2$`A`=12π`r`2. The length of the curved part of the perimeter is given by $L=\pi r$`L`=π`r`, and the total perimeter is given by $P=\pi r+2r$`P`=π`r`+2`r`.

A door that can be opened in both directions has left a semicircular mark on the floor. The door is $92$92 cm wide.

What are the dimensions of the rectangular mat, whose length $l$

`l`is greater than width $w$`w`, that must be purchased to most effectively protect the floor from scratches?$l=\editable{},w=\editable{}$

`l`=,`w`=

A lion has escaped through a hole in the fencing at an animal sanctuary. The wall that it escaped through runs east to west for over $50$50 km in both directions from the hole.

The animal handling team has to set up a perimeter to recapture the lion. They know it could only have travelled at most $21$21 km in the time since it escaped.

What length of fencing will they require to set up a semicircular perimeter? Round your answer to two decimal places.

How much area will they have to search? Round your answer to two decimal places.

A printed circuit has a circular component that the printer must go around. The printer recognises the bottom left corner of the circuit as the origin.

What would be the equation of a semicircle that connects the points $\left(6,7\right)$(6,7) and $\left(12,7\right)$(12,7) that are either side of the circular component?

Is there another semi-circle that can connect the two points?

Yes

ANo

BYes

ANo

B

Apply the geometry of conic sections

Apply the geometry of conic sections in solving problems