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?
ReplyDeleteI know his paternal grand daughter who lives in Sydney. She is the wife of my brother in law. I heard from her about late Khalifa. I as a BUET student in 1976 encountered his equation in our co-ordinate geometry course. Khalifa was a gold medalist from Calcutta University. In the beginning during the British period he worked as a magistrate. He became a teacher of Dhaka Varsity Math department when he settled in Dhaka after 1947 partition. It's true he hailed from Kushtia.
ReplyDeleteHow did you find out the centre and radii of the circle x2+ y2 – 12x + 4y + 13 = 0
ReplyDeleteEq should be x2+ y2 – 12x + 4y + 15 = 0
ReplyDelete(x−6)
Delete2
+(y+2)
2
=25