Here we are going to see some example problems to know how to find equation of a circle in general form.
Standard equation of a circle with centre (0, 0) and radius r :
x2 + y2 = r2
Equation of a circle with centre (h, k) and radius r :
(x - h)2 + (y - k)2 = r2
General form of the equation of a circle :
x2 + y2 + 2gx + 2fy + c = 0
Example 1 :
Obtain the equation of the circles with radius 5 cm and touching x-axis at the origin in general form.
Solution :
Equation of a circle with centre (h, k) and radius r :
(x - h)2 + (y - k)2 = r2
The center lies on y axis. The center point will be at (0, 5) and (0, -5).
Center (0, 5) and r = 5 (x - h)2 + (y - k)2 = r2 (x - 0)2 + (y - 5)2 = 52 x2 + y2 - 10y + 25 - 25 = 0 x2 + y2 - 10y = 0 |
Center (0, -5) and r = 5 (x - h)2 + (y - k)2 = r2 (x - 0)2 + (y + 5)2 = 52 x2 + y2 + 10y + 25 - 25 = 0 x2 + y2 + 10y = 0 |
So, the required equations are
x2 + y2 - 10y = 0 and x2 + y2 + 10y = 0
Example 2 :
Find the equation of the circle with centre (2, -1) and passing through the point (3, 6) in general form.
Solution :
Equation of a circle with centre (h, k) and radius r :
(x - h)2 + (y - k)2 = r2
Centre (h, k) ==> (2, -1).
(x - 2)2 + (y + 1)2 = r2 -----(1)
The given circle is passing through the point (3, 6).
Then, substitute 3 for x and 6 for y.
(3 - 2)2 + (6 + 1)2 = r2
12 + 72 = r2
1 + 49 = r2
50 = r2
Then,
(1)-----> (x - 2)2 + (y + 1)2 = 50
x2 - 2(x)(2) + 22 + y2 + 2(y)(1) + 12 = 50
x2 - 4x + 4 + y2 + 2y + 1 = 50
x2 + y2 - 4x + 2y + 5 = 50
Subtract 50 from each side.
x2 + y2 - 4x + 2y - 45 = 0
Example 3 :
Find the equation of circles that touch both the axes and pass through (-4, -2) in general form
Solution :
The center point will be at (-r, -r)
By applying the point passes through the circle and center, we get
(x - h)2 + (y - k)2 = r2
(-4 + r)2 + (-2 + r)2 = r2
16 + r2 - 8r + 4 - 4r + r2 - r2 = 0
20 + r2 - 12r = 0
r2 - 12r + 20 = 0
(r - 10) (r - 2) = 0
r = 10 and r = 2
Equation of a circle center is at (-10, -10) and radius is 10.
(x + 10)2 + (y + 10)2 = 102
x2 + 20x + 100 + y2 + 20y + 100 - 100 = 0
x2 + 20x + y2 + 20y + 100 = 0
x2 + y2 + 20x + 20y + 100 = 0
Equation of a circle center is at (-2, -2) and radius is 2.
(x + 2)2 + (y + 2)2 = 22
x2 + 4x + 4 + y2 + 4y + 4 - 4 = 0
x2 + 4x + y2 + 4y + 4 = 0
x2 + y2 + 4x + 4y + 4 = 0
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