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+y2−2xy−4x−4y−4=0
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B
x2+y2−2xy+4x−4y−4=0
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C
x2+y2+2xy−4x+4y−4=0
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D
x2+y2+2xy−4x−4y+4=0
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Solution
The correct option is Cx2+y2+2xy−4x+4y−4=0 The 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