There exists two circles passing through (1,-1) and touching the lines x+y=2 and x-y=2 whose radii are r1 and r2 and r1+r2=a√2,then a is equal to

Asked by Shambhu Nath Pandey | 6th Oct, 2016, 10:14: PM

Expert Answer:

Let space open parentheses straight x minus straight h close parentheses squared space space plus space space open parentheses straight y minus straight k close parentheses squared space equals space straight r squared.
Centre space lies space on space either space of space bisectors space of space angle space between space the space tangents space which space are comma
fraction numerator straight x plus straight y minus 2 over denominator square root of 2 end fraction equals plus-or-minus fraction numerator straight x minus straight y minus 2 over denominator square root of 2 end fraction space space or space space straight y equals space 0 space space or space straight x space equals space 2
therefore space space straight k space equals space 0 space or space space straight h space equals space 2

Again space by space applying space straight p space equals space straight r space with space any space tangent
fraction numerator straight h plus straight k minus 2 over denominator square root of 2 end fraction equals straight r
Put space straight k space equals space 0 space
therefore open parentheses straight h minus 2 close parentheses squared equals 2 straight r squared................ left parenthesis straight i right parenthesis

Circle space passes space through space the space point space left parenthesis 1 comma space minus 1 right parenthesis
therefore open parentheses 1 minus straight h close parentheses squared space space plus space space open parentheses 1 plus straight k close parentheses squared space equals space straight r squared............ left parenthesis ii right parenthesis
Put space straight k space equals space 0 space in space left parenthesis ii right parenthesis
straight h squared minus 2 straight h plus 1 plus 1 space equals space straight r squared
therefore straight h squared minus 2 straight h plus 2 space equals open parentheses straight h minus 2 close parentheses squared over 2
rightwards double arrow space straight h space equals space 0
rightwards double arrow space straight r space squared equals space 4 over 2 equals 2

Equation space of space the space circle space is space straight x squared plus straight y squared space equals space 2


Put space straight h space equals space 2 space we space get space the space quadrtic space equation space in space straight k space as
straight k squared plus 4 straight k plus 4 equals 0
rightwards double arrow straight k space equals space minus 2
rightwards double arrow space straight r space squared equals space 4 over 2 equals 2

Equation space of space the space circle space is space open parentheses straight x minus 2 close parentheses squared plus open parentheses straight y plus 2 close parentheses squared space equals space 2



Now space as space we space know space equations space of space both space the space circles space so space its space easy space to space find space the space value space of space straight a.
 
 
 
 

Answered by Vijaykumar Wani | 7th Oct, 2016, 04:20: PM