If the line x + 2by + 7 =0 is a diameter of the circle circle x^2+y^2-6x+2y =0, then find the value of b.
Answers
Given :
The line equation x + 2 b y + 7 = 0 is a diameter of circle
The circle equation is x² + y² - 6 x + 2 y = 0
To Find :
The value of b
Solution :
We know,
The standard equation of circle is x² + y² + 2 g x + 2 f y + c = 0
And Given circle equation is x² + y² - 6 x + 2 y = 0
Comparing we get
2 g = - 6 and 2 f = 2
i.e g = i,e f =
Or, g = - 3 Or, f = 1
As , The center of any circle is ( - g , - f )
So, From g and f value , center of this circle is ( - ( - 3 ) , - 1 ) = ( 3 , - 1 )
Again
The diameter equation is x + 2 b y + 7 = 0 , This diameter must pass through center ( 3 , - 1 )
Satisfying this center on diameter equation
i.e x + 2 b y + 7 = 0
Or, 3 + 2 b ( - 1 ) + 7 = 0
Or, 3 - 2 b + 7 = 0
Or, 10 - 2 b = 0
∴ b =
i.e b = 5
So, The value of b = 5