Exercises
You already know the circle from the previous page. The challenge now is to recognise it quickly, from an equation or from a statement, and to solve the whole problem.
The circle: standard and general form
The circle in standard and general form
The standard form is $(x-h)^2+(y-k)^2=r^2$, with centre $(h, k)$ and radius $r$. Expanding it gives the general form $x^2+y^2+2gx+2fy+c=0$, with centre $(-g, -f)$ and radius $r=\sqrt{g^2+f^2-c}$. This is the form you will find in the Formulae and Tables booklet.
To write the equation from the centre and a point on the circle, find $r$ as the distance between them, then use the standard form. Expand the brackets to get the general form.
Determine the coordinates of the centre and the radius of the circle with general equation $x^2 + y^2 - 8x + 10y - 8 = 0$.
Determine the coordinates of the centre and the radius of the circle with general equation $x^2 + y^2 - 8x + 10y - 8 = 0$.
A circle has its centre at point $C(-3, 5)$ and passes through the origin. Write its standard equation and its general equation.
A circle has its centre at point $C(-3, 5)$ and passes through the origin. Write its standard equation and its general equation.
Which circle is it? Look at the statement
When there is no equation, look at which condition the problem describes. "Same distance from a fixed point" is a circle. Three points not on a line also determine exactly one circle.
Find the equation of the perpendicular bisector of the line segment with endpoints $A(2, -3)$ and $B(-4, 5)$.
Find the equation of the perpendicular bisector of the line segment with endpoints $A(2, -3)$ and $B(-4, 5)$.
An underwater drone moves in such a way that its distance to a fixed buoy at $A(-2, 0)$ is always double its distance to another fixed buoy at $B(4, 0)$. Find the…
An underwater drone moves in such a way that its distance to a fixed buoy at $A(-2, 0)$ is always double its distance to another fixed buoy at $B(4, 0)$. Find the equation of the drone's path, simplify it, and identify the type of geometric curve it describes.
Multi-step context problems
Real-life problems rarely give you the centre and radius directly: first translate the statement into those data.
Write the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 12 = 0$ and $s: 5x + 12y - 10 = 0$.
Write the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 12 = 0$ and $s: 5x + 12y - 10 = 0$.
Three seismic stations detect an earthquake at coordinates $A(1, 4)$, $B(-3, 2)$ and $C(5, -2)$ (in kilometres). Assuming that the seismic waves travel at the same speed…
Three seismic stations detect an earthquake at coordinates $A(1, 4)$, $B(-3, 2)$ and $C(5, -2)$ (in kilometres). Assuming that the seismic waves travel at the same speed and arrive at the same time at all three stations, find the coordinates of the epicentre and the radius of the affected area (distance to the stations).
A marine radar has a detection range modelled by the circle $x^2 + y^2 - 4x + 6y - 12 \leq 0$. A ship follows a straight-line path given by the equation…
A marine radar has a detection range modelled by the circle $x^2 + y^2 - 4x + 6y - 12 \leq 0$. A ship follows a straight-line path given by the equation $3x + 4y + k = 0$. Determine the values of the parameter $k$ for which the ship is detected by the radar at some point along its path.
Conics and lines: intersections and tangents
To find the points where a line meets a circle, you substitute the line (solved for one variable) into the equation of the conic: you get a quadratic equation whose discriminant decides whether there are $2$, $1$ or $0$ solutions.
Does the line meet the circle at all?
You can also compare the perpendicular distance $d$ from the centre to the line with the radius $r$. If $d
Tangent from an exterior point
To find the tangents to a circle from an exterior point $P$, write the line through $P$ as $y-y_1=m(x-x_1)$ and set the distance from the centre to the line equal to the radius (or set the discriminant to zero after substituting). Solve for $m$: two values give the two tangents. If you get only one value of $m$, the other tangent is the vertical line $x=x_1$, which this form cannot describe, so always check it separately.
Radical axis
For two intersecting circles $C_1=0$ and $C_2=0$ (both in the form $x^2+y^2+\dots=0$), subtracting the equations gives the line through their points of intersection, called the radical axis. Substitute it into one circle to find those points.
Find the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Graph the situation, showing both loci and the points of…
Find the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Graph the situation, showing both loci and the points of intersection.
You can do it on paper and send a photo with your answer.
From a control station located at point $P(0, 7)$, laser signals are sent to a satellite whose circular orbit at that instant is given by the equation…
From a control station located at point $P(0, 7)$, laser signals are sent to a satellite whose circular orbit at that instant is given by the equation $x^2 + y^2 - 4x + 2y - 15 = 0$. Find the equations of the two straight-line paths tangent to the orbit from the control station.
To find the tangent lines to the circle $C: x^2 + y^2 - 2x - 4y - 4 = 0$ drawn from the exterior point $P(5, 5)$, two methodological approaches are proposed: Method A…
To find the tangent lines to the circle $C: x^2 + y^2 - 2x - 4y - 4 = 0$ drawn from the exterior point $P(5, 5)$, two methodological approaches are proposed:
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Method A (General algebraic): Set up the family of lines passing through $P$, $y - 5 = m(x - 5)$, substitute into the conic, and set the discriminant of the resulting quadratic equation to zero ($\Delta = 0$).
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Method B (Metric geometric): Obtain the centre $C$ and the radius $R$ of the circle by completing squares, and set the distance from the centre to the general line $mx - y + (5 - 5m) = 0$ equal to the radius ($d(C, r) = R$).
- a) Solve the problem by fully applying both methods to obtain the values of the slopes $m$, and write the general equations of the two resulting tangents.
- b) Critically compare both methods: which of the two turns out to be notably more agile and with a lower computational burden in this case? Could you apply Method B with the same simplicity if the conic were an ellipse of the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$? Justify your answer based on the geometric properties of both curves.