Two circles are said to be orthogonal circles, if the tangent at their point of intersection are at right angles.
If two circles are cut orthogonally then it must satisfy the following condition.
2 g1 g2 + 2 f1 f2 = c1 + c2
General equation of circle :
x2 + y2 + 2gx + 2fy + c = 0
Example 1 :
Show that the circles
x2 + y2 - 8x + 6y - 23 = 0
and
x2 + y2 - 2x - 5y + 16 = 0
are orthogonal.
Solution :
From 1st equation :
x2 + y2 - 8x + 6y - 23 = 0
x2 + y2 + 2g1x + 2f1y + c1 = 0
2g1 = -8, g1 = -4
2f1 = 6, f1 = 3
and c1 = -2
From 2nd equation :
x2 + y2 - 2x - 5y + 16 = 0
2g2 = -2, g2 = -1
2f2 = -5, f2 = -5/2
and c2 = 16
Condition to prove two circles are orthogonal :
2 g1 g2 + 2 f1 f2 = c1 + c2
2(-4)(-1)+2(3)(-5/2) = -23+16
8-15 = -7
- 7 = -7
The two circles cut orthogonally and hence they are orthogonal circles.
Example 2 :
Show that the circles
x2 + y2 - 8x + 6y + 21 = 0
and
x2 + y2 - 2y - 15 = 0
are orthogonal.
Solution :
From 1st equation :
x2 + y2 - 8x + 6y + 21 = 0
x2 + y2 + 2g1x + 2f1y + c1 = 0
2g1 = -8, g1 = -4
2f1 = 6, f1 = 3
and c1 = 21
From 2nd equation :
x2 + y2 - 2y - 15 = 0
x2 + y2 + 2g2x + 2f2y + c2 = 0
2g1 = 0, g1 = 0
2f1 = -2, f1 = -1
and c1 = -15
2 (-4) (0) + 2 (-3) (-1) = 21 - 15
Condition to prove two circles are orthogonal :
2 g1 g2 + 2 f1 f2 = c1 + c2
2(-4)(0)+2(-3)(-1) = 21-15
0+6 = 6
6 = 6
The two circles cut orthogonally and hence they are orthogonal circles.
Example 3 :
Find the equation of the circle which passes through the point (1, 2) and cuts orthogonally each of the circles
x2 + y2 = 9
and
x2 + y2 − 2x + 8y − 7 = 0
Solution :
Let the required equation of the circle be
x2 + y2 + 2gx + 2fy + c = 0 ------(1)
The point (1, 2) lies on the circle
1+4+2g+4f+c = 0
2g+4f+c = −5 -----(2)
The circle (1) cuts the circle x2 + y2 = 9 orthogonally.
2g1g2 + 2f1f2 = c1 + c2
2g(0) + 2f(0) = c − 9
c = 9 ---(3)
Again the circle (1) cuts
x2 + y2 − 2x + 8y − 7 = 0
orthogonally.
2g(− 1) + 2f(4) = c − 7
-2g + 8f = 9 − 7 = 2
−g + 4f = 1 -----(4)
(2) becomes 2g + 4f = − 14
g + 2f = − 7 ------(5)
(4) + (5) => 6f = − 6 ⇒ f = − 1
(5) => g − 2 = − 7 => g = − 5
The required equation of the circle is
x2 + y2 − 10x − 2y + 9 = 0
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