Find the equations of the circle whose centre is (3,-1) and which cut off an intercept of length 6 from the line 2x-5y+18=0.

Asked by Anindya | 27th Sep, 2014, 10:49: AM

Expert Answer:

T h e space g e n e r a l space e q u a t i o ln space o f space c i r c l e space i n space s tan d a r d space f o r m space i s open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared comma space w h e r e comma space open parentheses h comma k close parentheses space a r e space t h e space c o o r d i n a t e s space o f space t h e space c e n t r e space o f space t h e space c i r c l e space a n d space r comma space t h e space r a d i u s space o f space t h e c i r c ; e. G i v e n space t h a t space left parenthesis 3 comma minus 1 right parenthesis space a r e space t h e space c o o r d i n a t e s space o f space t h e space c e n t r e space a n d space h e n c e space t h e space e q u a t i o n space o f t h e space c i r c l e space i s open parentheses x minus 3 close parentheses squared plus open parentheses y plus 1 close parentheses squared equals r squared A l s o space g i v e n space t h a t space t h e space l i n e space 2 x minus 5 y plus 18 equals 0 space c u t s space o f f space a n space i n t e r c e p t space o f space l e n g t h space 6. W e space k n o w space t h a t space t h e space l i n e space p a s sin g space t h r o u g h space t h e space c e n t r e space i s space a space p e r p e n d i c u l a r space b i s e c t o r space o f space t h e space c h o r d. L e t space u s space u s e space t h e space f o l l o w i n g space f o r m u l a space t o space f i n d space t h e space p e r p e n d i u c l a r space d i s tan c e space f r o m space t h e space c e n t r e space t o space t h e space c h o r d. d equals fraction numerator open vertical bar a x subscript C plus b y subscript C plus c close vertical bar over denominator square root of a squared plus b squared end root end fraction T h u s comma space d equals fraction numerator open vertical bar 2 cross times 3 minus 5 cross times open parentheses minus 1 close parentheses plus 18 close vertical bar over denominator square root of 2 squared plus open parentheses minus 5 close parentheses squared end root end fraction rightwards double arrow d equals fraction numerator open vertical bar 6 plus 5 plus 18 close vertical bar over denominator square root of 4 plus 25 end root end fraction rightwards double arrow d equals fraction numerator open vertical bar 29 close vertical bar over denominator square root of 29 end fraction equals square root of 29 C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
S i n c e space t h e space l i n e space p a s sin g space t h r o u g h space t h e space c e n t r e space b i s e c t s space t h e space c h o r d comma space w e space h a v e comma A M equals M B equals 6 over 2 equals 3 space u n i t s A l s o comma space O M perpendicular A M C o n s i d e r space t h e space r i g h t space t r i a n g l e space O A M. T h u s comma space b y space a p p l y i n g space P y t h a g o r a s space T h e o r e m comma space w e space h a v e comma O A squared equals A M squared plus O M squared rightwards double arrow O A squared equals 3 squared plus open parentheses square root of 29 close parentheses squared rightwards double arrow O A squared equals 9 plus 29 rightwards double arrow O A squared equals 38 rightwards double arrow O A equals open parentheses square root of 38 close parentheses T h u s comma space t h e space e q u a t i o n space o f space t h e space c i r c l e space i s open parentheses x minus 3 close parentheses squared plus open parentheses y plus 1 close parentheses squared equals open parentheses square root of 38 close parentheses squared

Answered by Vimala Ramamurthy | 29th Sep, 2014, 08:54: AM