A VARIABLE CHORD PQ OF THE PARABOLA Y=X^2 SUBTENDS A RIGHT ANGLE AT THE VERTEX, THEN THE LOCUS OF POINTS OF INTERSECTION OF THE NORMALS AT P & Q IS

Asked by Sunil Soni | 14th Oct, 2015, 10:48: PM

Expert Answer:

F o r space t h e space p a r a b o l a space y equals x squared comma space s l o p e space o f space t a n g e n t equals fraction numerator d y over denominator d x end fraction equals 2 x. space H e n c e comma space s l o p e space o f space n o r m a l space i s space minus fraction numerator 1 over denominator 2 x end fraction. L e t space t w o space p o i n t s space P space a n d space Q space o f space t h e space c h o r d space h a s space c o o r d i n a t e s space open parentheses x subscript 1 comma space y subscript 1 close parentheses space a n d space space open parentheses x subscript 2 comma space y subscript 2 close parentheses. T h e space c o o r d i n a t e s space o f space v e r t e x space f o r space t h e space g i v e n space p a r a b o l a space a r e space O left parenthesis 0 comma space 0 right parenthesis. S l o p e space o f space O P space equals space y subscript 1 over x subscript 1 comma space s l o p e space o f space O Q equals y subscript 2 over x subscript 2. G i v e n space t h a t space O P perpendicular O Q. space H e n c e comma space y subscript 1 over x subscript 1 cross times y subscript 2 over x subscript 2 equals negative 1 rightwards double arrow y subscript 1 y subscript 2 equals negative x subscript 1 x subscript 2..................................... left parenthesis 1 right parenthesis y subscript 1 equals x subscript 1 squared space a n d space y subscript 2 equals x subscript 2 squared........... left parenthesis 2 right parenthesis F r o m space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space minus x subscript 1 x subscript 2 equals x subscript 1 squared x subscript 2 squared rightwards double arrow x subscript 1 x subscript 2 open parentheses 1 plus x subscript 1 x subscript 2 close parentheses equals 0 rightwards double arrow x subscript 1 x subscript 2 equals negative 1 E q u a t i o n space o f space n o r m a l space p a s s i n g space t h r o u g h space P space i s space y minus y subscript 1 equals negative fraction numerator 1 over denominator 2 x subscript 1 end fraction open parentheses x minus x subscript 1 close parentheses...... left parenthesis 3 right parenthesis E q u a t i o n space o f space n o r m a l space p a s s i n g space t h r o u g h space Q space i s space y minus y subscript 2 equals negative fraction numerator 1 over denominator 2 x subscript 2 end fraction open parentheses x minus x subscript 2 close parentheses....... left parenthesis 4 right parenthesis S u b t r a c t i n g space left parenthesis 4 right parenthesis space f r o m space left parenthesis 3 right parenthesis comma space w e space g e t space y subscript 2 minus y subscript 1 equals x over 2 open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses rightwards double arrow 1 over x equals fraction numerator 1 over denominator 2 open parentheses space y subscript 2 minus y subscript 1 close parentheses end fraction open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses rightwards double arrow 1 over x equals fraction numerator 1 over denominator 2 open parentheses x subscript 2 squared minus x subscript 1 squared close parentheses end fraction open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses rightwards double arrow 1 over x equals negative fraction numerator 1 over denominator 2 open parentheses x subscript 1 plus x subscript 2 close parentheses x subscript 1 x subscript 2 end fraction equals fraction numerator 1 over denominator 2 open parentheses x subscript 1 plus x subscript 2 close parentheses end fraction.... left parenthesis 5 right parenthesis N o w space m u l t i p l y i n g space left parenthesis 3 right parenthesis space b y space 1 over x subscript 2 space a n d space left parenthesis 4 right parenthesis space b y space 1 over x subscript 1 space a n d space s u b t r a c t i n g space w e space g e t y open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses plus y subscript 2 over x subscript 1 minus y subscript 1 over x subscript 2 equals 1 half open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses rightwards double arrow y open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses plus fraction numerator y subscript 2 x subscript 2 minus y subscript 1 x subscript 1 over denominator x subscript 1 x subscript 2 end fraction equals 1 half open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses rightwards double arrow y open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses plus fraction numerator x subscript 2 cubed minus x subscript 1 cubed over denominator x subscript 1 x subscript 2 end fraction equals 1 half open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses rightwards double arrow y open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses minus open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses open parentheses x subscript 1 squared plus x subscript 2 squared plus x subscript 1 x subscript 2 close parentheses equals 1 half open parentheses 1 over x subscript 2 minus 1 over x subscript 1 close parentheses rightwards double arrow y minus open parentheses open parentheses x subscript 1 plus x subscript 2 close parentheses squared minus x subscript 1 x subscript 2 close parentheses equals 1 half rightwards double arrow y minus open parentheses x squared over 4 plus 1 close parentheses equals 1 half........ u s i n g space left parenthesis 5 right parenthesis rightwards double arrow y minus 3 over 2 equals 1 fourth x squared w h i c h space i s space t h e space t h e space r e q u i r e d space l o c u s.

Answered by satyajit samal | 15th Oct, 2015, 12:33: PM