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If (-1,-1) is the focus and x+y+4=0 is the directrix of a parabola, then its vertex is

Options

  1. A(- 3 2 ,- 3 2 )
  2. B(- 5 2 ,- 5 2 )
  3. C(- 1 4 ,- 1 4 )
  4. D( 1 4 , 1 4 )

Correct answer

A. (- 3 2 ,- 3 2 )

Step-by-step solution

Let P(x, y) be any point on the parabola whose focus is S(-1,-1) and directrix is x+y+4=0 Let M be foot of perpendicular drawn from P to the directrix. Then, array lc & S P=P M (S P)^2=(P M)^2 (x+1)^2+(y+1)^2= ( |x+y+4| (1)^2+(1)^2 )^2 & 2 (x^2+1+2 x+y^2+1+2 y ) & (x^2+y^2+16+2 x y+8 y+8 x ) & . 2 x^2+y^2+2 x+2 y+2 ) & =x^2+y^2+2 x y+8 y+8 x+16 & x^2+y^2-4 x-4 y-2 x y-12=0 & (x-y)^2-4 x-4 y-12=0 & (x-y)^2=-4(x+y+3) Let & x-y=X, x+y+3=Y & X^2=-4 Y array On comparing with it X^2=-4 A Y , we get A=1 Since, vertex of E

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