Q: If secants are drawn through P(2,-1) to the circle x^2+y^2+2x = 0 meeting it at A and B and Q lies on segment AB thenA) Locus of Q such that PA,PQ,PB are in H.P is 3x - y + 2 = 0B) Locus of Q such that PA,PQ,PB are in A.P is x^2 + y^2 - x + y - 2 = 0C) Locus of Q such that PA,PQ,PB are in G.P is x^2 + y^2 - 4x + 2y - 4 = 0D) If a is least and b is greatest and aPQb then G.M of a and b is 3

Asked by Sunil Soni | 21st Sep, 2015, 10:32: PM

Expert Answer:

A right parenthesis space T h e space p o l a r space e q u a t i o n space o f space t h e space c i r c l e space w i t h space r e s p e c t space t o space t h e space p o l e space P space a n d space f i x e d space l i n e space P C space i s r squared minus 2 d r cos theta plus d squared minus R squared equals 0. W h e r e space apostrophe d apostrophe space i s space t h e space d i s tan c e space b e t w e e n space p o l e space a n d space t h e space c e n t e r space o f space t h e space c i r c l e. R space i s space t h e space r a d i u s space o f space t h e space c i r c l e. E q u a t i o n space o f space t h e space c i r c l e space i n space s tan d a r d space c a r t e s i a n space f o r m space i s space open parentheses x plus 1 close parentheses squared plus y squared equals 1. space p o l e equals left parenthesis 2 comma space minus 1 right parenthesis right parenthesis H e n c e comma space d equals square root of 10 comma space R equals 1 P o l a r space e q u a t i o n space b e c o m e s space r squared minus 2 cross times square root of 10 cross times r c o s theta plus 9 equals 0 T h e space a b o v e space i s space a space q u a d r a t i c space e q u a t i o n space w h o s e space r o o t s space g i v e apostrophe r apostrophe space v a l u e s space o f space t w o space p o i n t s space o n space t h e space c i r c l e space f o r space a n y space s e c a n t space p a s sin g space t h r o u g h space P. H e n c e comma space w e space g e t space f o r space P B space s e c a n t comma space P A plus P B equals 2 square root of 10 cos theta comma space P A. P B equals 9 G i v e n space t h a t space P A comma space P Q space a n d space P B space a r e space i n space H. P. rightwards double arrow fraction numerator 1 over denominator P A end fraction plus fraction numerator 1 over denominator P B end fraction equals fraction numerator 2 over denominator P Q end fraction rightwards double arrow fraction numerator P A plus P B over denominator P A. P B end fraction equals 2 over r rightwards double arrow fraction numerator 2 square root of 10 c o s theta over denominator 9 end fraction space equals 2 over r rightwards double arrow r cos theta equals fraction numerator 9 over denominator square root of 10 end fraction H e n c e comma space l o c u s space o f space apostrophe Q apostrophe space i s space a space s t r a i g h t space l i n e space p e r p e n d i c u l a r space t o space P C space a t space a space d i s tan c e space o f space fraction numerator 9 over denominator square root of 10 end fraction space f r o m space P. L e t space i n space c a r t e s i a n space c o o r d i n a t e s space i t s space e q u a t i o n space i s space y equals m x plus c S i n c e comma space i t apostrophe s space p e r p e n d i c u l a r space t o space P C comma space m equals 3 S o space e q u a t i o n space b e c o m e s space y equals 3 x plus c. A l s o comma space i t space i s space a t space a space d i s tan c e space o f space fraction numerator 9 over denominator square root of 10 end fraction space f r o m space P. H e n c e comma space open vertical bar fraction numerator 3 cross times 2 minus open parentheses negative 1 close parentheses plus c over denominator square root of 10 end fraction close vertical bar equals fraction numerator 9 over denominator square root of 10 end fraction open vertical bar 7 plus c close vertical bar equals 9 rightwards double arrow c equals 2 space left parenthesis r e j e c t i n g space c equals negative 16 space b e c a u s e space w e space g e t space a space l i n e space o u t s i d e space t h e space g i v e n space c i r c l e right parenthesis H e n c e comma space L o c u s space o f space apostrophe Q apostrophe space i s space y equals 3 x plus 2 rightwards double arrow 3 x minus y plus 2 equals 0  B right parenthesis space P A comma space P Q space a n d space P B space a r e space i n space A. P. rightwards double arrow P Q minus P A equals P B minus P Q rightwards double arrow Q A equals Q B H e n c e comma space Q space i s space t h e space m i d p o i n t space o f space apostrophe A apostrophe space a n d space apostrophe B apostrophe. L e t space t h e space c o o d i n a t e s space o f space Q space b e space left parenthesis h comma space k right parenthesis S i n c e space i t apostrophe s space t h e space m i d p o i n t space o f space t h e space c h o r d space A B comma space e q u a t i o n space o f space c h o r d space A B space i s S apostrophe equals T rightwards double arrow h squared plus k squared plus 2 h equals h x plus k y plus open parentheses x plus h close parentheses rightwards double arrow h squared plus k squared minus h x minus k y plus h minus x equals 0 S I n c e space P left parenthesis 2 comma space minus 1 right parenthesis space i s space a space p o i n t space o n space i t comma space w e space g e t h squared plus k squared minus 2 h plus k plus h minus 2 equals 0 rightwards double arrow h squared plus k squared minus h plus k minus 2 equals 0 H e n c e comma space l o c u s space o f space Q space i s space x squared plus y squared minus x plus y minus 2 equals 0  C right parenthesis space P A comma space P Q space a n d space P B space a r e space i n space G. P. rightwards double arrow P A. P B equals P Q squared rightwards double arrow 3 squared equals P Q squared space left parenthesis 3 space i s space t h e space l e n g t h space o f space tan g e n t space f r o m space P space t o space t h e space g i v e n space c i r c l e right parenthesis rightwards double arrow 9 equals open parentheses h minus 2 close parentheses squared plus open parentheses k plus 1 close parentheses squared rightwards double arrow h squared plus k squared minus 4 h plus 2 k minus 4 equals 0 H e n c e comma space l o c u s space o f space Q space i s x squared plus y squared minus 4 x plus 2 y minus 4 equals 0  D right parenthesis space s t a t e m e n t space i n space o p t i o n space D space i s space n o t space c l e a r. O p t i o n s space A comma space B space a n d space C space a r e space c o r r e c t.

Answered by satyajit samal | 24th Sep, 2015, 05:47: AM

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