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Question

Let tangent drawn to the parabola C1:y2=4ax at P meets the y-axis at Q. Another parabola C2 with vertex Q and focus (0,0) is drawn which passes through (2a,0). Identify which of the following statements can be CORRECT?

A
The equation of the curve C2 is x2=4a(y+a)
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B
The equation of the curve C2 is x2=4a(y+a)
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C
Point P is the extremity of latus rectum of curve C1
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D
Possible equation of directrix of curve C2 is y2a=0
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Solution

The correct option is D Possible equation of directrix of curve C2 is y2a=0
Tangent at P(at2,2at) is
ty=x+at2
Q(0,at)

Now, equation of parabola with vertex (0,at) and (0,0) as its focus is x2=4at(yat)
The curve passes through (2a,0) 4a2=4at(at)
t2=1t=±1

Possible equation of curve C2 is
x2=4a(ya)
or x2=4a(y+a)
Co-ordinates of P are (a,±2a)
[ Extremities of latus rectum of C1]

Possible equation of directrix of C2:
ya=ay2a=0
or y+a+a=0y+2a=0

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