Formula
1) To find the equation of any 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 given then
x² + y² = r² (r is radius of the circle)
EXERCISE-A
1) Find the equation of the circle whose
A) centre is (4,5), radius is 7. x²+ y² - 8x - 10y - 8= 0
B) centre (a,b), radius √(a²+ b²). x²+ y² - 2ax - 12by= 0.
C) Centre (a cos k, a sin k), radius a. x²+ y² - (2a cos k)x - (2a sin k)y = 0.
D) A) centre is (a,a), radius √2 a. x²+ y² - 2ax - 2ay = 0
2) Find the centre and radius of followings:
A) (x -1)² + y² = 4. (1,0); 2
B) (x +5)² + (y+1)² = 8. (-5,-1); 3
C) x²+ y² - 4x + 6y = 5. (2,-3), 3√2
D) x²+ y² - x + 2y - 3= 0. (1/2, -1), √17/2
E) x² + (y +2)²= 9. (0,-2), 3
3) If the equations of the two diameters of a circle are x - y = 5 and 2x+ y = 4 and the radius of the circle is 5, find the equation of the circle. x²+ y² - 6x + 4y - 12= 0
4) If the equations of the two diameters of a circle are 2x +4y = 5 and 2x- 3y = -12 and area is 154 square units, find the equation of the circle. (x+3)²+ (y-2)²- 49= 0
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* General Equation Of A circle:
x²+ y² + 2gx + 2fy + c = 0.
Where Centre=(-g, -f)
Radius= √(g²+ f²- c)
Note -1
The equation x²+ y² + 2gx + 2fy + c = 0 represents a circle of radius √(g²+ f²- c).
A) If g²+ f²- c> 0 then the radius of the circle is real and hence the circle is also real.
B) if g²+ f²- c = 0 then the radius of the circle is zero, Such a circle is known as a point circle.
C) If g²+ f²- c < 0, then the radius of √(g²+ f²- c) of the circle is imaginary but the centre is a real. Such a circle is called an imaginary circle as it is not possible to draw a such a circle.
Note-2
Special features of the general equation x²+ y² + 2gx + 2fy + c = 0 of the circle are:
A) it is quadratic in both x and y.
B) coefficient of x²= coefficient of y².
In solving problem it is advisable to keep the coefficient of x² and y² unity.
C) there is no term containing xy i.e., the coefficient xy is zero.
D) it contains three arbitrary constants viz, g, f and c.
Note-3
The equation ax²+ ay² + 2gx + 2fy + c = 0, a≠ 0 also represents a circle. This equation can also be written as
x²+ y² + 2gx/a + 2fy/a + c/a = 0,
The coordinates of the centre are (-g/a, -f/a) and,
Radius= √(g²/a² + f²/a² -c/a).
Note-4
On comparing the general equation x²+ y² + 2gx + 2fy + c = 0 of a circle with the general equation of second degree ax²+ 2hxy+ b y² + 2gx + 2fy + c = 0 we find that it represents a circle if a= b i.e., coefficient of x²= coefficient of y² and h= 0 i.e., coefficient of xy = 0.
EXERCISE -B
1) Find the centre and the radius of the circle
a) x²+ y² -6x + 4y -12 = 0. (3,-2),5
b) x²+ y² +6x -8y -24 = 0. (-3,4),7
c) 2x²+ 2y² -3x + 5y = 7 (3/4,-5/4), 3√10/4
d) 1/2(x²+ y²) +x cos ¢+ y sin¢ -4 = 0. (-cos¢,- sin¢), 3
e) x²+ y² - ax - by = 0. (a/2,b/2), 1/2 √(a²+ b²)
2) Find the equation of circle that passes through the points
A) (1,0),(-1,0),(0,1). x²+ y² = 1
B) (5,-8),(2,-9),(2,1). x²+ y² -4x +8y = 5
C) (5,7),(8,1),(1,3). 3(x²+ y²)- 29x -19y = -56
D) (0,0),(-2,1),(-3,2). x²+ y² - 3x -11y =0
3) Find the equation of the circle whose centre is at the point (4,5) and which passes through the centre of the circle x²+ y² - 6x +4y = 12.
4) Find the question the circle passing through (1,0) and (0,1) having the smallest possible radius. x²+ y² - x -y =0
5) Show that the points
A) (9,1),(7,9)(- 2,12) and (6,10)
B) (3,-2),(1,0)(-1,-2) and (1,-4)
C) (5,5),(6,4)(-2,4) and (7,1)
are concyclic
6) a) Find the equation of the circle which passes through the point (1,-2) and (4,-3) and has its centre on the line 3x +4y= 7. 15(x²+ y²) - 94x +18y +55=0.
b) Find the equation of the circle which passes through the point (3,2) and (-2,0) and has its centre on the line 2x -y= 3. x²+ y²+ 3x +12y +2=0.
c) Find the equation of the circle which passes through the point (3,7) and (5,5) and has its centre on the line x- 4y= 1. x²+ y²+ 6x +2y -90=0.
7) Find the equation of the circle which circumscribes the Triangle formed by the lines:
A) x+ y+ 3=0, x -y+ 1=0, x- 3=0. x²+ y²- 6x +2y - 15=0.
B) 2x+ y- 3=0, x -y- 1=0, 3x+ 2y- 5=0. x²+ y²- 13x -5y +16 =0.
C) x+ y- 3=0, 3x - 4y=6, x- y=0. x²+ y²+ 4x +6y - 12=0.
D) x+ y- 6=0, x +y=4, x+ 2y=5. x²+ y²- 17x -19y +50=0.
8) Prove that the centres of the three circles x²+ y²- 4x - 6y - 12=0, x²+ y²+2x +4y - 10=0, and x²+ y²- 10x -16y - 1=0 are collinear.
9) Prove that the radius of the circles x²+ y²=1, x²+ y²-2x -6y - 6=0, and x²+ y²- 4x -12y - 9=0 are in AP.
10) Find the equation of the circle concentric with the circle
A) 2x²+ 2y²+8x + 10y - 39 =0, having its area equal to 16π square units. 4x²+ 4y²+ 16x +20y - 23=0
B) x²+ y²-6x +12y +15=0 and double of its area. x²+ y²-6x +12y - 15=0
C) x²+ y²- 4x -6y - 3=0 and which touches the y-axis. x²+ y²-4x -6y +9=0
11) Find the radius of the circle (x- cos¢+ y sin¢ - a)²+ (x sin¢ -y cos¢ - b)²=k². (0,0) √(a²+ b²)
12) Find the area of an equilateral triangle inscribed in the circle x²+ y²+ 2gx +2fy + c=0. 3√3/4 (g²+f²-c)
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Formula
Equation of a circle is (x - h)² + (y - k)² = a².
1) When the centre of the circle coincides with the origin i e., h= k = 0. Then
Equation is x²+ y² = a² (a is radius)
2) When the circle passes through the origin then
x² + y² - 2hx - 2ky = 0.
3) When the circle touches x-axis then a = k
Equation is x² + y² - 2hx - 2ay + h²= 0.
4) When the circle touches y-axis then h = a
Equation is x² + y² - 2ax - 2ky + k²= 0.
5) When the circle touches both the axes then a = k = h
Equation is x² + y² - 2ax - 2ay + a²= 0.
6) When the circle passes through the origin and centre lies on x-axis. then a = h and k= 0
Equation is x² + y² - 2ax = 0.
7) When the circle passes through the origin and centre lies on y-axis. then a = k and h= 0
Equation is x² + y² - 2ay = 0.
8)
EXERCISE -C
1) Find the equation of the circle whose centre is (1,2) and which passes through the point (4,6). x²+ y²- 2x - 4y - 20 =0.
2) Find the equation of a circle whose centre lies on the positive direction of y-axis at a distance of 6 from the origin and whose radius is 4. x²+ y²-12y+20=0
3) Find the equation of a circle:
a) which touches both the axes at a distance of 6 units from the origin. x²+ y²- -12x -12y+36=0
b) which touches y-axis at a distance of 5 units from the origin and radius 6 units. x²+ y²- -10x -12y+25=0
c) which touches both the axes and passes through the point (2,1). x²+ y²- 2x -2y+1=0, x²+ y²- 10x -10y+25=0
d) Passing through the origin, and radius 17 and ordinate of the centre is -15. x²+ y² ±16x +30y =0
e) Find the equation of a circle of radius 5 whose centre lies on x-axis and passes through the point (2,3). x² + y² +4x -21= 0.
f) Centre is (0, -4) and which touches the x-axis. x²+ y² + 8y= 0
g) Centre is (3, 4) and which touches the x-axis. x²+ y² - 6x- 8y+9= 0
h) the x-axis at the origin and radius is 5. x²+ y²- 10y= 0
4) a) Find the equation of a circle which touches y-axis at a distance of 4 units from the origin and cuts an intercept of 6 units along the positive direction of x-axis. x²+ y² - 10x ± 8y+16= 0
b)Which passes through the origin and cuts off intercepts of length 'a', each form positive direction of the axes. x²+ y² - ax - ay = 0
c) Find the equation of the circle which touches the lines x=0, y=0 and x = a. (x- a/2)²+ (y± a/2)²= (a/2)²
5) Find the equation of the circle whose centre is (2,3) and which touches the c-axis. x²+y²-4x -6y +4 = 0
6) Find the equation of the circle which touches both the axes and has a radius 3 units. x²+y²-6x -6y +9 = 0
7)
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EXERCISE -D
1) Find the equation of the circle in each of the following cases.
a) Centre (0,0) & radius √2. x²+y²= 2
b) Centre (2,3)& radius √13. x²+y²+4x - 6y= 0
c) Centre(3,-2) and radius 5. x²+y² - 6x -12 y= 0
d) Centre (a cos¢, a sin¢) and radius a. x²+y² -2ax cos¢ - 2ay sin¢= 0
2) Find the question of the circle whose diameter has the end points
a) (-2,2) and (2,-2). x²+y² - 6x = 8
b) (1/2,2) and (-1,0). 2(x²+y²) +x -4y -1 = 0
c) (3,-1) and (3,1). x²+y² - 6x +8= 0
3) Which of the following equation represents a circle ?
a) x²+y² +4= 0.
b) 2(x²+y²) -8x +4y +11 = 0.
c) x²+y² +4(x +y) -3 = 0.
4) Find the position of the following points relative to the corresponding circle:
A) (2,1); x²+y² -4x +4y -6 = 0. In
B) (-1,1); x²+y² -2x +4y +1 = 0. Out
C) (3,2); x²+y² -3x +4y -12 = 0. On
D) Show that the point (3,4) lie on the circle x²+y²- 6x +2y -15 = 0.
5) Prove that the radius of three circles is equations are
A) x²+y²= 1, x²+y²+ 6x -2y = 6, and x²+y²-12x +4y -9 = 0 are in AP.
B) x²+y²= 1, x²+y²-2x +2y = 7, and x²+y²-6x -2y -71 = 0 are in GP.
6) Find the parametric equation of the following circles:
A) x²+y²= 16. x=4 cos t, y= 4 sin t
B) x²+y²- 6x +4y = 12. x=3+ 5 cos t, y= -2+ 5 sin t
C) x²+y²-x +2y +1 = 0 x=1/2 + 1/2 cos t, y= -1+ 1/2 sin t
D) x²+y²+ax +by = 0 x=-a/2 + √(a²+ b²)cos t, y= -b/2 + √(a²+ b²)sin t
7) Find the value of k, if the circle x²+y²+4x -7y -k = 0 has diameter of 9 units. 4
8) Find the Cartesian equations of the following circles:
a) x= 4 cos ¢, y= x= 4 sin ¢. x²+y²= 16
b) x= -2+7 cos ¢, y= 2+ 7 sin ¢. x²+y² + 4x - 4y -33=0
c) x= h+r cos ¢, y= k+ r sin ¢. (x- h)²+(y- k)²=r²
EXERCISE-E
1) a) Find the equation of the circle whose centre is (2,3) and which passes the point (4,5). x²+y²- 4x -6y +5 = 0
b) Find the equation of the circle whose centre is (-2,-5) and which passes through the centre of the circle 3x²+3y²+6x -9y +16 = 0. 4(x²+y²)+ 16x +40y-57= 0
2)a) Find the equation of the circle is radius whose radius is √3 and which is concentric with the circle x²+y²- 3x +5y -1 = 0. 4x²+4y²-12x +20y +22 = 0
b) Find the equation of the circle which is concentrated with the circle x²+y²- 2x +4y -5 = 0 and which passes through the centre of the circle x²+y²+2x -4y +5 = 0. x²+y²- 2x +4y -15 = 0
c) Show that the two circles x²+y²-4x +6y +5 = 0 and x²+y²- 4x +6y -11 = 0 are concentric.
d) Find the values of m and n, if the two circles x²+y²-2x +4y -2 = 0 and x²+y²+ mx +ny -7= 0 are concentric. -2,4
e) Find the equation of the circle which is concentric with the circle x²+y²-2x -6y -e = 0 and passes through the point of intersection of the lines 2x+ 3y= 1 and x+ y= 1. x²+y²-2x -6y -7= 0
3) Find the radius and centre of the circle if extreme of the diameter are:
A) (4-6) and (-2,-2). (1,-4), √13
B) (7,9) and (-1-3). (3,3), 2√13
4) Show that the point (3,4) lie on the circle x²+y²-6x +2y -15 = 0.
5) Find the value of K, if the circle x²+y²+ 4x - 7y -k = 0 has diameter of 9 units. 4
6) a) Find the equation of the circle which passes through the points (3,2) and (5,4) and whose centre is on the line 3x+ 2y -12= 0. x²+y²+4x -18y +11 = 0
b) Find the equation of the circle which passes through the points (3,-1) and (0,4) and whose centre lies on the x-axis. x²+y²+2x -16 = 0
c) Find the equation of the circle which passes through the points (0,3) and (0,-7) and whose centre lies on the y-axis. x²+y²+4y -21 = 0
d) Find the value of c for which the length of the diameter of the circle x² +y² -8x +4y +c = 0 is 10. Find also the equation of the diameter of the circle which is parallel to the line 3x- 2y +1= 0. -5, 3x - 2y -16= 0
e) Find the value of c for which the diameter of the circle x² +y² +10x +cy +4 = 0 is 10. Find also the equation of the diameter of the circle which is perpendicular to the line 2x- 3y -1= 0. ±4, 3x + 2y +19= 0, or 3x+ 2y+11= 0
7) Find the value of K, for which the following points are concyclic (2,3),(0,2),(4,5) and (0,K). 2 or 7
Exercise -G
1) a) Prove that the circles x²+y²-4x +6y +8 = 0 and x²+y²-10x - 6y +14 =0 touch each other externally.
b) Prove that the circles x²+y²-2x +4y = 0 and x²+y²-10x +20 =0 touch each other externally.
c) Prove that the circles x²+y² = 2 and x²+y²- 6(x +y)+10 =0 touch one another. Find the point of contact. (1,1)
d) Prove that the circles x²+y²-4x +6y +8 = 0 and x²+y²-10x - 6y +14 =0 touch at the point (3,-1)
e) If two circles x²+y²+2ax + c²= 0 and x²+y²+ 2by +c² =0 touch each other externally., Prove that 1/a² + 1/b²= 1/c².
f) Prove that the circles x²+y²+ 2gx +2fy = 0 and x²+y²+ 2g'x +2f'y =0 touch each other if f'g = g'f.
EXERCISE - H
1) find the equation of the common chord of the two circles x²+y²-2x -4y+ 6y -36= 0 and x²+y²-5x +8y -43 = 0. x-2y+7= 0
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