Question

# Consider the parabola whose focus at (0,0) and tangent at vertex is x−y+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+2xy−4x+4y−4=0The distance between the focus and the tangent at the vertex is |0−0+1|√12+12=1√2 The directrix is the line parallel to the tangent at vertex and at a distance 2×1√2 from the focus. Let the equation of the directirx be, x−y+λ=0 So, ∣∣ ∣∣λ√12+12∣∣ ∣∣=2√2 ⇒λ=2 Let P(x,y) be any moving point on the parabola. Then, OP=PM x2+y2=(x−y+2√12+12)2 ⇒2x2+2y2=(x−y+2)2 ⇒x2+y2+2xy−4x+4y−4=0

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