Q47 / 120 · VITEEE 2018 · Mathematics

The focus of the curve $y^{2}+4 x-6 y+13=0$ is

  • A
    $(2,3)$
  • Correct answer
    B
    $(-2,3)$
  • C
    $(2,-3)$
  • D
    $(-2,-3)$

Solution

The given equation of curve is: $$ y^{2}+4 x-6 y+13=0 $$ which can be written as: $\mathrm{y}^{2}-6 \mathrm{y}+9+4 \mathrm{x}$ $$ \begin{array}{l} +4=0 \\ \Rightarrow\left(y^{2}-6 y+9\right)=-4(x+1) \\ \Rightarrow(y-3)^{2}=-4(x+1) \end{array} $$ Put $Y=y-3$ and $X=x+1$ On comparing $Y^{2}=4 a X$ Length of focus from vertex, $a=-1$ At focus $X=a$ and $Y=0 \Rightarrow x+1=-1$ $$ \begin{array}{l} \Rightarrow \mathrm{x}=-2 \\ \therefore \mathrm{y}-3=0 \Rightarrow \mathrm{y}=3 \\ \therefore \quad \text { Focus is }(-2,3) \end{array} $$

Hence, the correct answer is option: 'B'.

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