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Question

Consider the parabola whose focus at (0,0) and tangent at vertex is xy+1=0.
The equation of the parabola is

A
x2+y22xy4x4y4=0
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B
x2+y22xy+4x4y4=0
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C
x2+y2+2xy4x+4y4=0
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D
x2+y2+2xy4x4y+4=0
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Solution

The correct option is C x2+y2+2xy4x+4y4=0
The distance between the focus and the tangent at the vertex is |00+1|12+12=12
The directrix is the line parallel to the tangent at vertex and at a distance 2×12 from the focus.
Let the equation of the directirx be,
xy+λ=0
So,
∣ ∣λ12+12∣ ∣=22
λ=2
Let P(x,y) be any moving point on the parabola. Then,
OP=PM
x2+y2=(xy+212+12)2
2x2+2y2=(xy+2)2
x2+y2+2xy4x+4y4=0

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