A circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that plane is always constant.
Formula :
1) To find the equation of an circle whose centre and radius are given.
(x- h)²+ (y - k)²= a²
The above equation is known as the central form of the equation of a circle.
2) if the centre of the circle is at the origin and radius is a, then equation of circle = x²+ y² = a²
EXERCISE -1
1) Find the equation of a circle whose
A) centre is (2,-3) and radius 5. x²+ y² - 4x + 6y - 12= 0
B) centre is at origin and radius 6. x²+ y² - 36= 0
C) centre is (2,-3) and radius 4. (x+2)²+ (y -3)² = 16
D) centre is (a,b) and radius √(a²+ b²). x²+ y² - 2ax - 2by = 0
E) centre is (0, -1) and radius 1. x²+ y² + 2y = 0
F) centre is (a cos k, a sin k) and radius a. x²+ y² - (2a cos k)x -(2a sin k) y = 0
G) centre is (a,a) and radius a √2. x²+ y² - 2ax - 2ay = 0
H) centre is at the centre and radius 4. x²+ y² - 16= 0
2) Find the centre and radius of each of the following circles:
A) x²+ (y+2)²= 9. (0,-2), 3
B) x²+ y²- 4x+ 6y - 12= 0. (2,-3),5
C) (x+1)²+ (y- 1)²= 4. (-1,1), 2
D) x²+ y²+ 6x- 4y +4= 0. (-3, 2),3
E) (x -1)²+ y² - 4= 0. (1,0),2
F) (x+5)²+ (y+1)² - 9= 0. (-5,-1),3
G) x²+ y²- 4x+ 6y - 5= 0. (2,-3),3√2
H) x²+ y²- x+ 2y - 3= 0. (1/2,-1), √17/2
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EXERCISE -2
1) Find the equation of a circle whose
A) centre is (2,-1) and which passes through the point (3,6). x²+ y²- 4x+ 2y - 45= 0
B) centre is (2,-5) and which passes through the point (3,2). x²+ y²- 4x+ 10y - 21= 0
C) centre is (1,2) and which passes through the point (4,6). x²+ y² - 2x - 4y - 20= 0
2)A) Find the equation of a circle passing through the point (2,4) and having its centre at the intersection of the lines x - y= 4 and 2x+ 3y+7= 0. x²+ y²- 2x+ 6y - 40= 0
B) Find the equation of a circle whose centre is (2,-5) and which is passing through the intersection of the lines 3x + 2y= 11 and 2x+ 3y+7= 4. x²+ y²- 4x+ 6y +3= 0
C) Find the equation of a circle passing through the point (-1,3) and having its centre at the intersection of the lines x - 2y= 4 and 2x+ 5y+1= 0. x²+ y²- 4x+ 2y - 20= 0
D) Find the equation of the circle passing through the point of intersection of the lines x+ 3y= 0 and 2x- 7y= 0 and whose centre is the point of intersection of the lines x+ y + 1= 0 and x -2y+ 4= 0. x²+ y²+ 4x- 2y = 0
E) If the equations of two diameters of a circle are 2x+ y= 6 and 3x+ 2y = 4 and the radius is 10, find the equation of the circle. x²+ y²- 16x+ 20y + 64 = 0.
F) If the equations of two diameters of a circle are x- y= 5 and 2x+ y = 4 and the radius is 5, find the equation of the circle. x²+ y²- 6x+ 4y -12 = 0.
G) If the equations of two diameters of a circle are 2x - 3y+ 12= 0 and x+ 4y = 5 and the area is 154 square units. find the equation of the circle. (x+3)²+ (y-2)² = 49
H) Find the equation of the circle which has its centre at the point (3,4) and touches the the line 5x+ 12y -1= 0. 169(x²+ y²- 6x - 8y)+ 381 = 0
I) If the line 2x- y+1= 0 touches the circle at the point (2,5) and the centre of the circle lies on the line x+ y= 9. Find the equation of the circle. (x -6)²+ (y-3)²= 20.
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SOME PARTICULAR CASES
The equation of a circle with centre at (h, k) and radius equal to a, is
(x- h)²+ (y - k)²= a²
I) When the centre of the circle coincides with the origin then
h= k = 0
Then, the equation is x²+ y² = a²
II) When the circle passes through the origin then the equation is
x²+ y² - 2hx - 2ky = 0
III) When the circle touches x-axis. It means a= k
x²+ y² - 2hx - 2ay + h² = 0
IV) When the circle touches y-axis. It means h= a
x²+ y² - 2ax - 2ky + k² = 0.
V) When the circle touches both the axes: It means h= k= a
x²+ y² - 2ax - 2ay + a² = 0
VI) When the circle passes through the origin and centre lies on x-axis. It means k= 0, h = a.
x²+ y² - 2ax = 0
VII) When the circle passes through the origin and centre lies on y-axis. It means h= 0, k = a.
x²+ y² - 2ay = 0.
EXERCISE -3
1) Find the equation of the circle which touches:
A) the x-axis and whose centre is (3,4). x²+ y²- 6x - 8y + 9 = 0
) the x-axis at the origin and whose radius is 5. x²+ y² - 10y = 0
) both the axis and whose is 5. x²+ y²± 10x ± 10y + 25 = 0
) The lines x= 0, y= 0 and x= a. (X- a/2(² + (y ± a/2)²= (a/2)²
)
2) Find the equation of the circle which passes through two points on the x-axis which are at distances 4 from the origin and whose radius is 5. x²+ y²± 6y - 16 = 0
3) Find the equation of the circle which passes through the origin and cuts off intercepts 3 and 4 from the positive parts of the axes respectively. (x- 3/2)²+ (y- 2)² = (5/2)²
4)
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