The circle
The coverage area of a wifi antenna, the range of a radar, the (almost) circular orbit of a satellite: all of them are described by the simplest of the four curves, the one you already know. Now you give it an equation and use it to solve problems involving lines and other circles.
The standard equation
Write the equation of the circle describing the coverage area of a Wi-Fi antenna, knowing that the antenna is located at point $(2, -3)$ on a Cartesian plane (measured…
Write the equation of the circle describing the coverage area of a Wi-Fi antenna, knowing that the antenna is located at point $(2, -3)$ on a Cartesian plane (measured in kilometres) and has a maximum range of $5$ km.
Determine the coordinates of the centre and the length of the radius of the circular area watered by an agricultural sprinkler whose boundary is given by the equation…
Determine the coordinates of the centre and the length of the radius of the circular area watered by an agricultural sprinkler whose boundary is given by the equation $(x-4)^2+(y+1)^2=36$.
The general equation
If you expand the standard equation you get a form without brackets:
To go back: the centre is $C\left(-\dfrac{D}{2},-\dfrac{E}{2}\right)$ and the radius is $r=\sqrt{\left(\dfrac{D}{2}\right)^2+\left(\dfrac{E}{2}\right)^2-F}$.
Find the centre and radius of the circle describing the outer edge of a gear in a computer-aided design blueprint, whose general equation is $x^2+y^2-6x+8y=0$.
Find the centre and radius of the circle describing the outer edge of a gear in a computer-aided design blueprint, whose general equation is $x^2+y^2-6x+8y=0$.
Determine whether the equation $2x^2+2y^2+4x-8y+20=0$ mathematically represents a valid circle in a video game physics engine. Justify your answer by calculating its…
Determine whether the equation $2x^2+2y^2+4x-8y+20=0$ mathematically represents a valid circle in a video game physics engine. Justify your answer by calculating its centre and the value of its radius squared.
Position of a point and of a line
A point can lie inside, outside or on the circle, depending on how $d(P,C)$ compares with $r$. A line can be a secant (two points in common), a tangent (one point, and perpendicular to the radius) or exterior (none).
To study a line relative to the circle, you substitute the line into the equation and look at the discriminant of the resulting quadratic equation: positive (secant), zero (tangent) or negative (exterior).
Study analytically the relative position between the orbit of an artificial satellite, modelled by the circle $x^2+y^2=4$, and the straight-line trajectory of an…
Study analytically the relative position between the orbit of an artificial satellite, modelled by the circle $x^2+y^2=4$, and the straight-line trajectory of an asteroid given by the equation $y=x+2$. Determine the points of intersection if they exist.
Calculate the equation of the tangent line to the circular trajectory traced by a hammer thrower at the exact moment they release the wire. The trajectory of the hammer…
Calculate the equation of the tangent line to the circular trajectory traced by a hammer thrower at the exact moment they release the wire. The trajectory of the hammer is given by the circle $x^2+y^2=25$ and the athlete releases the wire when the ball passes through the point $(3, 4)$.
Two circles relative to each other
Two circles can be exterior, tangent (internally or externally), intersecting (they meet at two points) or one can lie inside the other without touching. The key is to compare the distance between the centres with the sum and the difference of the radii.
Study the relative position between the wavefronts generated by two drops falling into a pond, modelled by the circles $C_1: x^2+y^2=9$ and $C_2: (x-4)^2+y^2=4$.
Study the relative position between the wavefronts generated by two drops falling into a pond, modelled by the circles $C_1: x^2+y^2=9$ and $C_2: (x-4)^2+y^2=4$.
The GPS system of a mobile phone receives signals from two telecommunications towers. The first tower is at the origin of coordinates and estimates that the mobile phone…
The GPS system of a mobile phone receives signals from two telecommunications towers. The first tower is at the origin of coordinates and estimates that the mobile phone is $5$ km away. The second tower is at the point $(5, 0)$ and estimates that the mobile phone is $4$ km away. Find the exact coordinates of the two possible points where the mobile phone could be located by calculating the intersection of both circles.