Center of a circle C lies on the line 2x - 2y + 9 = 0 and it cuts the circle x squared plus y squared space equals space 4 orthogonally
The Circle C passes through two fixed points whose sum of
 
a) abscissa is -9/2                   b) abscissa is 9/2
c) ordinates is  9/2                  c) ordinates is -9/2
 
 
 

Asked by Sunil Soni | 10th Aug, 2016, 05:30: PM

Expert Answer:

Let space equation space of space the space circle space be space straight x squared plus straight y squared plus 2 gx plus 2 fy plus straight c equals 0.
Its space centre space open parentheses negative straight g space comma space minus straight f close parentheses space lies space on space 2 straight x space minus space 2 straight y space plus space 9 space equals 0
rightwards double arrow negative 2 straight g space plus space 2 straight f space plus space 9 space equals 0.
It space cuts space orthogonally space the space circle space straight x squared plus straight y squared minus 4 equals 0.
2 straight g subscript 1 straight g subscript 2 plus 2 straight f subscript 1 straight f subscript 2 equals straight c subscript 1 plus straight c subscript 2
0 plus 0 equals straight c minus 4
straight c equals 4
Putting space the space values space of space straight c space and space straight g space we space get space the space equation space of space the space circle space as
straight x squared plus straight y squared plus left parenthesis 9 plus 2 straight f right parenthesis straight x plus 2 fy plus 4 equals 0
or open parentheses space straight x squared plus straight y squared plus 9 straight x plus 4 right parenthesis plus 2 straight f left parenthesis straight x plus straight y close parentheses equals 0
Above space equation space is space of space the space form space straight S space plus space λP space equals space 0
Above space represents space family space of space circles space which space passes space through space the space points space of space intersection.
rightwards double arrow space straight x squared plus straight y squared plus 9 straight x plus 4 space equals space 0 space and space straight x space plus space straight y space equals space 0

Solving space above space equation space we space get comma
two space fixed space points space as space open parentheses negative 1 half comma space 1 half close parentheses space space space and space left parenthesis negative 4 comma space 4 right parenthesis.

Option space left parenthesis straight a right parenthesis space and space left parenthesis straight b right parenthesis space are space correct.

Answered by Vijaykumar Wani | 12th Aug, 2016, 05:06: PM