question_answer

The equation of the parabola whose vertex is at\[(2,-1)\]and focus at\[(2,-3),\]is:

A)  \[{{x}^{2}}-4x-8y-12=0\]

B)         \[{{x}^{2}}-4x+8y+12=0\]

C)         \[{{x}^{2}}+8y=12\]

D)         \[{{x}^{2}}-4x+12=0\]

E)         none of the above

Correct Answer: B

Solution :

Let co-ordinate of directrix in which meet the axis at Z is \[(\alpha ,\beta )\] \[\therefore \]  \[\frac{\alpha +2}{2}=2\]\[\Rightarrow \]\[\alpha =2\] and        \[\frac{\alpha -3}{2}=-1\Rightarrow \beta =1\] \[\therefore \]Co-ordinate of Z is (2, 1). \[\therefore \]equation of directrix is\[y=1\] \[\therefore \]  \[PS=PM\] \[\Rightarrow \]               \[{{(x-2)}^{2}}+{{(y+3)}^{2}}={{(y-1)}^{2}}\] \[\Rightarrow \]               \[{{x}^{2}}-4x+8y+12=0\]