Find the equation of the following circles: A circle has its centre on the line x + y = 1 and passes through the origin and the point (4,2).

flatantsmu 2022-09-05 Answered
Find the equation of the following circles: A circle has its centre on the line x + y = 1 and passes through the origin and the point ( 4 , 2 ).
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Answers (2)

recepiamsb
Answered 2022-09-06 Author has 9 answers
Given any two points on a circle the center lies on to the perpendicular bisector of the chord between them. Here the perpendicular bisector passes through the midpoint ( 1 / 2 ) ( ( 0 , 0 ) + ( 4 , 2 ) ) = ( 2 , 1 ). The slope of the chord is clearly + 1 / 2 so, by the "negative reciprocal rule" the perpendicular line has slope 2. So the center lies on
y 1 = 2 ( x 2 ) , y = 2 x + 5
Since the center also lies on y = 1 x we then have for the center:
1 x = 2 x + 5 , x = 4 , y = 3
meaning the center is ( 4 , 3 ). The radius should now be easy to figure out given that ( 4 , 2 ) is on the circle centered at ( 4 , 3 ), and the equation of the circle follows.
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Rohan Mcpherson
Answered 2022-09-07 Author has 1 answers
Hint
Let the center be C ( c , 1 c )
If r is the radius,
r 2 = c 2 + ( 1 c ) 2 = ( c 4 ) 2 + ( 1 c 2 ) 2
The last equation will give us the value of c
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