The point on the line x-y+2=0 from which the
tangent to the parabola y^2=8x is perpendicular 
to the given line is (a,b),then the line ax+by+c=0 is:

Asked by Sunil Soni | 17th Oct, 2015, 10:07: PM

Expert Answer:

begin mathsize 11px style left parenthesis straight a comma space straight b right parenthesis space is space straight a space point space on space straight x minus straight y plus 2 equals 0 rightwards double arrow straight a minus straight b plus 2 equals 0 rightwards double arrow straight a minus straight b equals negative 2.... left parenthesis 1 right parenthesis  Any space line space perpendicular space to space straight x minus straight y plus 2 equals 0 space is space straight y equals negative straight x plus straight K Since comma space left parenthesis straight a comma straight b right parenthesis space lies space on space this space line space as space well comma space straight b equals negative straight a plus straight K rightwards double arrow straight K equals straight a plus straight b..... left parenthesis 2 right parenthesis The space combined space equation space of space tangent space and space the space parabola space straight y squared equals 8 straight x space is open parentheses negative straight x plus straight K close parentheses squared equals 8 straight x rightwards double arrow straight x squared minus straight x open parentheses 2 straight K plus 8 close parentheses plus straight K squared equals 0 For space straight a space tangent comma space this space quadratic space must space have space equal space roots. Hence space straight D space equals space open parentheses 2 straight K plus 8 close parentheses squared minus 4 straight K squared equals 0 rightwards double arrow 4 straight K squared plus 64 plus 32 straight K minus 4 straight K squared equals 0 rightwards double arrow 32 straight K equals negative 64 rightwards double arrow straight K equals negative 2... left parenthesis 3 right parenthesis  Using space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma space we space get straight a plus straight b equals negative 2 space and space straight a minus straight b equals negative 2 Hence comma space straight a equals negative 2 space and space straight b equals 0.  The space line space ax plus by plus straight c equals 0 space becomes space minus 2 straight x plus straight c equals 0 straight x equals straight c over 2 comma space straight a space line space parallel space to space straight Y minus axis. space end style

Answered by satyajit samal | 18th Oct, 2015, 12:21: PM