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) \[{{x}^{2}}-4x+12=0\]

Correct Answer: B

Solution :

Key Idea: If the \[x-\]coordinates of a vertex and focus are same, then the axis is parallel to \[y-\]axis. And if the y-coordinates of a vertex and focus are same, then the axis is parallel to \[x-\]axis. Given vertex is \[(2,\,\,-1)\] and focus\[(2,\,\,-3)\]. By the condition-of parabola                                                 \[P{{S}^{2}}=P{{M}^{2}}\] \[\Rightarrow \]               \[{{(x-2)}^{2}}+{{(y+3)}^{2}}={{\left( \frac{y-1}{\sqrt{1}} \right)}^{2}}\] \[\Rightarrow \]               \[{{x}^{2}}-4x+4+{{y}^{2}}+9+6y={{y}^{2}}+1-2y\] \[\Rightarrow \]               \[{{x}^{2}}-4x+8y+12=0\]