The vertex of the locus of a point that divides a chord of slope 2 of the parabola y 2 = 4x internally in the ratio 1 : 2 is
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Let 
and 
be exremities of the chord with slope 2.
∴ 
= 2 ⇒ t 1 + t 2 = 1 ...(i)
Let R (h, k) be co-ordinates of the point which divides PQ in the ratio 1:2 then
⇒ h = 
and k = 
or 3h = 
+ (1 – t 1 ) 2 and 3k = 4t 1 + 2(1 – t 1 ) {from equation (i)}
Eliminating t 1 , 3h = 3 
+ 1
or 9k 2 – 16k – 4h + 8 = 0
⇒ k 2 – 
= 0
⇒ 
∴ Locus of R(h, k) is 
which is a parabola with vertex 
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