A. R. Khalifa (Azizur Rahman Khalifa), Professor of Dhaka University, deduced a simple formula for determining an equation of a circle which passes through three (given) points. For this reason, the formula is also known as A. R. Khalifa's Formula.
Let two circles (larger & smaller circle) intersect each other at points A (x1,
y1) & B (x2, y2) respectively. Let C (x3,
y3) be any point on the larger circle. It is required to find the formula of the larger circle, that passes through the three given points A (x1,
y1), B (x2, y2) & C (x3, y3).
Now, we know that the equation of a circle drawn on the straight line joining the points A (x1,
y1) & B (x2, y2) as diameter is;
(x –
x1) (x – x2) + (y – y1) (y – y2) =
0 - - - - (1)
Let,
C
(x, y) = (x – x1) (x – x2) + (y – y1) (y – y2) - - - - (2)
Again, the equation of a straight line passing through the two given points A (x1, y1) & B (x2, y2) is;
=> (x – x1) (y1 – y2) = (x1 – x2) (y – y1)
=> (x –
x1) (y1 – y2) – (y – y1) (x1 – x2) = 0 - - - - (3)
Let,
L
(x, y) = (x – x1) (y1 – y2) – (y – y1) (x1
– x2) - - - - (4)
We also know that the equation of a circle passing through the point of intersection of a circle & a straight line is;
C (x, y) + K L (x, y) = 0 - - - - (5) [Where, K is a constant]
Equation (5) passes through the point C (x3,
y3), i.e.
C (x3,
y3) + K L (x3,
y3) = 0 [ K ≠ 0 ]
=> K L (x3, y3) = – C (x3, y3)
- - - - (6)
Putting the value of 'K' in equation (5), we get;
- - - - (7)
Equation (7) is the required formula with which the equation of a circle passing through three given points can be determined.
Example:
Problem: Find the equation of a circle passing through the points (2, 1), (10, 1) and (2, – 5). Find the radius & center of the circle.
Solution:
Here,
(x1,
y1) = (2, 1)
(x2,
y2) = (10, 1)
(x3,
y3) = (2, – 5)
Just replace the coordinates in the equation (7) with the given coordinates. After some deduction, the equation of the circle will become;
x2+ y2 – 12x + 4y + 13 = 0
Center of the circle is (6, – 2) & radius is = 5 unit (Q.E.D)
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The formula deduced by late A R Khalifa sir is very useful though it is difficult to memorise. The theme is more useful than the formula itself.
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DeleteI was studying for Admission and wanted to find methods which take less time.
ReplyDeleteMy math book featured AR Khlifa's law but there was no introduction who he was.
Now i am very proud to learn that he is from my country.Respect.
He's from daulatpur ,kushtia .
DeleteCan we write it as ......+k((y-y1)(x1-x2)-(x-x1)(y1-y2))=0
ReplyDeleteWe are proud for khalifa Azizur Rahaman proffesor Dhaka university,becase he was a man of Daulatpur,Kushtia.
ReplyDeleteAR Khalifa tutored me my ISc class mathematics in 1970. A great soul and a philanthropist.
ReplyDeleteHow did you know, A.R. Khalifa is or was a professor at DU?
ReplyDelete