In the parabola y 2 = 4ax, the locus of middle points of all chords of constant length c is
Text Solution
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Given |AB| = c
Let 
and 
If P(h, k) be the middle point of AB then
(h, k) = 
or 
and 
= t 1 + t 2
or 
= (t 1 + t 2 ) 2 – 2t 1 t 2 or 2t 1 t 2 = 
or t 1 t 2 = 
and t 1 + t 2 = 
since |AB| = c
or 
+ (2at 1 – 2at 2 ) 2 = c 2 ⇒ a 2 (t 1 – t 2 ) 2 {(t 1 + t 2 ) 2 + 4} = c 2
⇒ a 2 {(t 1 + t 2 ) 2 – 4t 1 t 2 }{(t 1 + t 2 ) 2 + 4} = c 2 ⇒ a 2 
= c 2
⇒ a 2 
= c 2 ⇒ (4ah – k 2 )(k 2 + 4a 2 ) = a 2 c 2
Hence locus of mid point of AB is (4ax – y 2 )(y 2 + 4a 2 ) = a 2 c 2
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