IF a BE THE ANGLE SUBTENDED AT THE FOCUS BY THE NORMAL CHORD AT THE POINT(w,w),w#0  ON 
THE PARABOLA y^2=4x, THEN EQUATION OF THE LINE THROUGH (1,2)AND MAKING AN ANGLE a WITH
THE x-AXIS IS:

Asked by Sunil Soni | 17th Oct, 2015, 10:23: PM

Expert Answer:

begin mathsize 11px style Slope space of space tangent space to space straight y squared equals 4 straight x space is space 2 over straight y Hence comma space slope space of space normal space is space minus straight y over 2. If space left parenthesis straight w comma space straight w right parenthesis space is space straight a space point space on space this space parabola comma space then space straight w squared equals 4 straight w rightwards double arrow straight w equals 4 space space left parenthesis because space straight w not equal to 0 right parenthesis Slope space of space normal space at space left parenthesis 4 comma space 4 right parenthesis space is space minus 4 over 2 equals negative 2 Equation space of space normal space at space left parenthesis 4 comma 4 right parenthesis space is space straight y minus 4 equals negative 2 open parentheses straight x minus 4 close parentheses rightwards double arrow straight y equals negative 2 straight x plus 12 To space find space the space coordinates space of space other space end space of space the space normal space chord comma form space combined space equation space of space the space normal space with space the space equation space of space parabola. open parentheses negative 2 straight x plus 12 close parentheses squared equals 4 straight x rightwards double arrow open parentheses straight x minus 6 close parentheses squared equals straight x rightwards double arrow straight x squared minus 13 straight x plus 36 equals 0 rightwards double arrow straight x subscript 1 plus straight x subscript 2 equals 13 One space end space of space the space normal space chord space has space straight X minus coordinate space 4. Hence comma space the space straight X minus coordinate space of space the space other space end space is space 9. straight Y space coodinate space of space this space point space is space found space by space substituting space straight X space value space in space equation space of space normal. straight y equals negative 2 cross times 9 plus 12 rightwards double arrow straight y equals negative 6 Hence comma space the space coordinates space of space two space ends space of space the space normal space chord space are left parenthesis 4 comma 4 right parenthesis space and space left parenthesis 9 comma space minus 6 right parenthesis. Focus space of space the space parabola space straight y squared equals 4 straight x space is space left parenthesis 1 comma space 0 right parenthesis Slope space of space line space joining space left parenthesis 4 comma 4 right parenthesis space and space left parenthesis 1 comma space 0 right parenthesis space is space straight m subscript 1 equals fraction numerator 4 minus 0 over denominator 4 minus 1 end fraction equals 4 over 3 Slope space of space line space joining space left parenthesis 9 comma space minus 6 right parenthesis space and space left parenthesis 1 comma space 0 right parenthesis space is space straight m subscript 2 equals fraction numerator negative 6 minus 0 over denominator 9 minus 1 end fraction equals negative 6 over 8 equals negative 3 over 4 Here space straight m subscript 1 straight m subscript 2 equals negative 1. space So comma space the space normal space chord space subtends space right space angle space at space focus. straight a equals 90 to the power of straight o Hence comma space euqation space of space straight a space line space passing space through space left parenthesis 1 comma 2 right parenthesis space and space making space 90 to the power of straight o space with space straight X minus axis space is straight x equals 1 space left parenthesis parallel space to space straight Y minus axis space passing space through space left parenthesis 1 comma space 2 right parenthesis right parenthesis end style

Answered by satyajit samal | 18th Oct, 2015, 01:35: PM