Concepts
Write the standard equation of a circle with centre (a,b) and radius r.
(x-a)2+(y-b)2=r2
Find the centre and radius.
(x-2)2+(y+3)2=16
Centre = (2, −3)
Radius = 4
How many intersection points can two circles have?
0, 1, or 2
What determines whether two circles intersect?
Distance between centres compared with radii.
What happens when
d = r1 + r2
Circles touch externally (1 intersection).
State the distance formula used to compare circle centres.
d=((x2-x1)2+(y2-y1)2)1/2
Expand
(x-3)2+(y+2)2=25
x2 - 6x + 9 + y2 + 4y + 4=25
Solve
x2+y2=25
(x-4)2+y2=9
(4,3),(4,-3)
Two circles have radii 5 and 4 units respectively. The distance between the centres of the circles is 10 units
Do they intersect?
No, because
10 > 5 + 4
Circles
x2+y2=25
(x-6)2+y2=13
Find intersection points.
(4,3), (4,−3)
What happens when
d < |r1 - r2|?
One circle lies completely inside the other with no intersection.
Write the equation of a circle with
Centre (3,-1)
Radius 5
(x-3)2 + (y+1)2 = 25
Find the intersection points of:
(x-1)2 + y2 = 10
(x+3)2 + y2 = 26
(1, (10)1/2), (1, -(10)1/2)
Two circles intersect at two points
Which inequality must be true?
|r1 - r2| < d < r1 + r2
Which of these cannot be an equation of a circle?
1) x² -35 + y² - 26x + 55y = 0
2) 3x² + 8y + 3y² -9x -23 = 0
3) 2x² + 8y + 2y² + 5x + 37 = 0
3)
Why do we subtract circle equations when solving intersections?
To eliminate x2 and y2 and obtain a linear equation.
Show that
x2+y2-4x+6y+9=0
represents a circle, and find its centre.
Centre = (2, −3)
Find intersection points.
(x-2)2+y2=13
(x+2)2+y2=13
(0,3),(0,-3)
Two circles intersect at right angles.
What relationship holds?
r12+r22=d2
The circle (x−5)² +(y−4)²=4 intersects a circle whose diameter has endpoints (4,3) and (8,5). Find the length of the common chord formed by the two circles.
4
How many solutions can a system of two circle equations have?
0, 1, or 2
Show that
x2+y2+2gx+2fy+c=0
represents a circle with centre
(-g,-f)
(Appropriate solution obvs)
Solve
x2+y2=10
(x-2)2+y2=2
(2,2),(2,-2)
Two circles have the same radius and intersect.
What is the locus of their intersection points?
The perpendicular bisector of the line joining centres.
How many circles with radius root41 have their centres on the parabola x2 - x + 2 and pass through the point
(1,-3)?
3