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 y−2a=0
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Solution
The correct option is D Possible equation of directrix of curve C2 is y−2a=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(y−at)
The curve passes through (2a,0)4a2=−4at(−at) ⇒t2=1⇒t=±1
Possible equation of curve C2 is x2=−4a(y−a)
or x2=4a(y+a)
Co-ordinates of P are (a,±2a) [ Extremities of latus rectum of C1]
Possible equation of directrix of C2: y−a=a⇒y−2a=0
or y+a+a=0⇒y+2a=0