Find the equation of the parabola whose focus is the point (2,3) and the directrix is the line x-4y+3 = 0. Also, find the lenght of its latus-rectum.

Asked by abhishek_nayak98 | 30th Jun, 2014, 06:53: PM

Expert Answer:

L e t space P left parenthesis x comma y right parenthesis space b e space a n y space p o i n t space o n space t h e space p a r a b o l a. G i v e n space t h a t space t h e space f o c u s space o f space t h e space p a r a b o l a space i s space S open parentheses 2 comma 3 close parentheses space a n d space t h e space e q u a t i o n space o f space t h e space d i r e c t r i x space i s space x minus 4 y plus 3 equals 0. D r a w space a space p e r p e n d i c u l a r space P M space f r o m space P space o n space t h e space d i r e c t r i x. T h e n space b y space d e f i n i t i o n comma space w e space h a v e S P equals P M rightwards double arrow S P squared equals P M squared rightwards double arrow open parentheses x minus 2 close parentheses squared plus open parentheses y minus 3 close parentheses squared equals open vertical bar fraction numerator x minus 4 y plus 3 over denominator square root of 1 squared plus open parentheses minus 4 close parentheses squared end root end fraction close vertical bar squared rightwards double arrow x squared minus 4 x plus 4 plus y squared minus 6 y plus 9 equals open parentheses x minus 4 y plus 3 close parentheses squared over 17 rightwards double arrow 17 open parentheses x squared plus y squared minus 4 x minus 6 y plus 13 close parentheses equals open parentheses x minus 4 y plus 3 close parentheses squared rightwards double arrow 17 x squared plus 17 y squared minus 68 x minus 102 y plus 221 equals x squared plus 16 y squared plus 9 minus 8 x y plus 6 x minus 24 y rightwards double arrow 16 x squared plus y squared plus 8 x y minus 74 x minus 78 y plus 212 equals 0 T h i s space i s space t h e space e q u a t i o n space o f space t h e space r e q u i r e d space p a r a b o l a.

L a t u s space r e c t u m space equals space 2 cross times open parentheses L e n g t h space o f space t h e space p e r p e n d i c u l a r space f r o m space t h e space f o c u s space o n space t h e space d i r e c t r i x close parentheses equals space 2 cross times open parentheses L e n g t h space o f space t h e space p e r p e n d i c u l a r space f r o m space open parentheses 2 comma 3 close parentheses space o n space x minus 4 y plus 3 close parentheses equals 2 cross times open vertical bar fraction numerator 2 minus 4 cross times 3 plus 3 over denominator square root of 1 squared plus 4 squared end root end fraction close vertical bar equals 2 cross times open vertical bar fraction numerator minus 7 over denominator square root of 1 plus 16 end root end fraction close vertical bar equals 2 cross times open vertical bar fraction numerator minus 7 over denominator square root of 17 end fraction close vertical bar L a t u s space R e c t u m equals fraction numerator 14 over denominator square root of 17 end fraction

Answered by Vimala Ramamurthy | 1st Jul, 2014, 09:39: AM