A variable straight line of slope 4 intersects the hyperbola 
at two points. The locus of the point which divides the line segment between these two points in the ratio 1 : 2 is
Text Solution
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Let 
be any point on the locus. Equation of the line through P and having slope 4 is 
.....(i)
Suppose this meets 
......(ii) in 
and
Eliminating y between (i) and (ii), we get 
⇒ ⇒ 
⇒ ⇒ 
......(iii)
This has two roots say 
; 
......(iv) and 
......(v)
Also, 
[ 
P divides AB in the ratio 1 : 2]
i.e., 
......(vi)
(vi) – (iv) gives, 
and
Putting in (v), we get
⇒ ⇒ 
⇒ ⇒ 
Required locus of 
is 
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