The centers of the circles touching both the coordinate axes lies on the line

x – y = 0 or x + y = 0 according to the quadrant in which the circle lies.

Based on this data answer the following 2 questions.

 

A) If the difference between the radii of the circles passing through (a, b) and touching the axes is C > 0 and a, b > 0 then the least value of a + b is

1 right parenthesis thin space space c space space space space space space space space space space space space space space space space 2 right parenthesis space square root of 2 c space space space space space space space space space space space 3 right parenthesis thin space fraction numerator c over denominator square root of 2 end fraction space space space space space space space space space 4 right parenthesis space 2 c

B) If one of the points of intersection of the two circles touching the axes is (p,q) then the length of the common chord is

1 right parenthesis thin space vertical line p plus-or-minus q vertical line space space space space space space space space space 2 right parenthesis space square root of 2 vertical line p plus-or-minus q vertical line space space space space space space space space space space space 3 right parenthesis space 2 vertical line p plus-or-minus q vertical line space space space space space space space space space space 4 right parenthesis space square root of 2 space vertical line p q vertical line

Asked by Sunil Soni | 10th Aug, 2016, 07:24: PM

Expert Answer:

The space circles space having space centres space on space the space line space straight x space minus space straight y space equals space 0 space space will space lie space either space on space first space or space third space quadrant.

The space circles space having space centres space on space the space line space straight x space plus space straight y space equals space 0 space space will space lie space either space on space second space or space fourth space quadrant.

Both space the space circle space passing space through space the space point space left parenthesis straight a comma space straight b right parenthesis.
Also space straight b greater than 0
As space the space circles space touches space the space axes space so space the space circles space lie space on space first space and space secod space qudrant space touching space each space other space at space point space left parenthesis straight a comma space straight b right parenthesis space on space straight y space axis.
rightwards double arrow straight a space equals space 0
therefore straight a space plus space straight b space equals 0 plus straight b space equals space straight b........... left parenthesis straight i right parenthesis

therefore The space centre space of space the space circle space on space the space line space straight x space minus space straight y space equals space 0 space space in space the space firat space quadrant space will space be space left parenthesis straight b space comma space straight b right parenthesis
and space space the space centre space of space the space circle space on space the space line space straight x space minus space straight y space equals space 0 space space in space the space firat space quadrant space will space be space left parenthesis negative straight b space comma space straight b right parenthesis.

space The space difference space between space the space radii space of space the space circles space is comma
straight C equals square root of left parenthesis straight b plus straight b right parenthesis squared plus left parenthesis straight b minus straight b right parenthesis squared end root equals square root of 2 straight b............. left parenthesis ii right parenthesis

space From space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis space we space get comma

straight a plus straight b space equals fraction numerator straight C over denominator square root of 2 end fraction
 
 
 
 
Please post each query seperately.

Answered by Vijaykumar Wani | 12th Aug, 2016, 04:52: PM