Circles: the general equation, two circles and loci · Grade 12 · 5 units
In Circles: the general equation, two circles and loci (Grade 12 · 5 units) you practise 5 skills: The general equation of a circle · The relative position of two circles · The common chord · Moving a circle and a tangent · Loci that are circles. Expanding the equation of a circle gives its general form: x squared plus y squared, plus terms in x and in y and a number, all equal to zero. To find the centre and the radius, complete the square: group the terms in x and the terms in y, and add the square of half of each coefficient to both sides. If the right side comes out positive, the figure is a circle and that number is the square of the radius. If it is zero, the figure is a single point, and if it is negative, there are no points at all.
The relative position of two circles depends on the distance between their centres. If it is greater than the sum of the radii, each circle is outside the other. If it equals the sum, the circles touch externally, and if it equals the difference, they touch internally. If it is between the difference and the sum, they intersect at two points, and if it is less than the difference, one circle is inside the other. Subtracting the equations of two intersecting circles cancels the squared terms and leaves the equation of a line: the line that the common chord lies on. When a circle moves, its radius stays the same and its centre moves. A tangent that moves with the circle stays parallel to itself, and the point where it touches moves just as the centre does. Some loci are circles. Label a general point by its coordinates, write the condition with the distance formula, square, expand and complete the square. For example, the points whose squared distances from two fixed points add up to a fixed number lie on a circle centred at the midpoint between them, if that number is large enough. The points twice as far from one point as from another also lie on a circle.
Tip: Before you take a radius from the general equation, check that the number on the right side is positive.
What you learn
The general equation of a circle
For example: The equation x² + y² + 10x − 12y + 57 = 0 describes a circle. What is the x coordinate of its centre?
Answer and solution
The answer: −5
Completing the square gives (x + 5)² + (y − 6)² = 4 . So the centre is (−5, 6) , and its x coordinate is −5 .
The relative position of two circles
For example: The circles (x − 4)² + (y + 3)² = 9 and (x + 5)² + (y + 3)² = 1 are given. What is their relative position?
- They touch externally
- They intersect at two points
- They touch internally
- Each circle is outside the other
Answer and solution
The answer: Each circle is outside the other
The centres are (4, −3) and (−5, −3) , so the square of the distance between them is d² = 81 . The radii are 3 and 1 , so (r₁ + r₂)² = 16 and (r₁ − r₂)² = 4 . So each circle is outside the other.
The common chord
For example: The circles (x + 3)² + (y + 5)² = 25 and (x + 4)² + (y + 7)² = 50 are given, and they intersect at two points. Which line passes through both points of intersection?
- 2x − y = −1
- x + 2y = 3
- 2x + y = −3
- x + 2y = −3
Answer and solution
The answer: x + 2y = −3
Subtracting the equations gives the line x + 2y = −3 .
Moving a circle and a tangent
For example: The circle (x + 4)² + (y − 5)² = 7 is given. It is moved one unit to the right and five units down. What is the equation of the circle that results?
- (x + 5)² + (y − 10)² = 7
- (x + 3)² + (y − 10)² = 7
- (x + 5)² + y² = 7
- (x + 3)² + y² = 7
Answer and solution
The answer: (x + 3)² + y² = 7
The centre (−4, 5) moves to (−3, 0) and the radius stays the same, so the equation is (x + 3)² + y² = 7 .
Loci that are circles
For example: The points A(6, −5) and B(0, 5) are given. What is the locus of the points P with PA² + PB² = 82 ?
- A line
- A single point
- No points at all
- A circle
Answer and solution
The answer: A circle
The midpoint of AB is N(3, 0) and AB² = 136 . Expanding and completing the square: PA² + PB² = 2 × PN² + 68 , so PN² = 7 . The square of the distance is positive, so the answer is a circle.
Not on the site yet
Parts of this topic that are not built yet: A locus that is an arc (an inscribed angle).
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).