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Question

A tangent is drawn to parabola y24x+4=0 at a point P which cuts the directrix at the point Q. If a point R is such that it divides QP externally in ratio 1:2, then the locus of point R is

A
y(x+1)2+4=0
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B
y2(x+1)+4=0
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C
y=0
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D
(x+1)(y1)24=0
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Solution

The correct option is B y2(x+1)+4=0

y24x+4=0y2=4(x1)
Therefore, we get
vertex : (1,0), Focus : (2,0) and Directrix : x=0
Let P be (1+t2,2t)
Slope of the tangent =1t
Equation of tangent at P is
y2t=1t(x(1+t2))y2t=xt1+t2t

Q lies on the directrix x=0
Coordinates of Q is (0,t1t) and P(1+t2,2t)
R divides PQ in 1:2 externally
R(x,y)=⎜ ⎜ ⎜ ⎜1+t21,2t2(t1t)1⎟ ⎟ ⎟ ⎟

x=1t2
and y=2tt=2y
x=14y2y2(x+1)+4=0

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