Consider the parabola 3y^2+4y-6x+8=0. The points on the axis of this parabola from where 3 distinct normals can be brawn are given by:

Asked by Sunil Soni | 17th Oct, 2015, 10:13: AM

Expert Answer:

3 y squared plus 4 y minus 6 x plus 8 equals 0 rightwards double arrow 3 open parentheses y squared plus 4 over 3 y close parentheses equals 6 x minus 8 rightwards double arrow 3 open parentheses y plus 2 over 3 close parentheses squared equals 6 x minus 8 plus 4 over 3 rightwards double arrow open parentheses y plus 2 over 3 close parentheses squared equals 6 over 3 open parentheses x minus 10 over 9 close parentheses rightwards double arrow open parentheses y plus 2 over 3 close parentheses squared equals 2 open parentheses x minus 10 over 9 close parentheses A x i s space o f space t h e space p a r a b o l a space i s space y equals negative 2 over 3 V e r t e x colon space open parentheses 10 over 9 comma space minus 2 over 3 close parentheses A n y space p o i n t space o n space t h e space a x i s space h a s space c o o r d i n a t e s space open parentheses h comma space minus 2 over 3 close parentheses rightwards double arrow Y squared equals 4 open parentheses 1 half close parentheses X space left parenthesis t r a n s f o r m i n g space t o space n e w space c o o r d i n a t e space a x e s space X comma Y right parenthesis L e t space a n y space p o i n t space o n space t h e space a x i s space a s space p e r space n e w space c o o r d i n a t e s space b e space left parenthesis H comma space 0 right parenthesis I t space w i l l space h a v e space t h r e e space n o r m a l s space p a s sin g space t h r o u g h space i t space i f space H greater than 2 a space left parenthesis p r o p e r t y right parenthesis rightwards double arrow H greater than space 2 open parentheses 1 half close parentheses rightwards double arrow H greater than 1 A s space p e r space o r i g i n a l space c o o r d i n a t e s comma space t h i s space i m p l i e s space h minus 10 over 9 greater than 1 rightwards double arrow h greater than 19 over 9 k equals negative 2 over 3 H e n c e comma space a l l space t h e space p o i n t s space o n space t h e space a x i s space w i t h space x minus c o o r d i n a t e greater than 19 over 9 w i l l space h a v e space t h r e e space d i s t i n c t space n o r m a l s space p a s sin g space t h r o u g h space i t.

Answered by satyajit samal | 18th Oct, 2015, 09:53: AM