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  4. The focus of the curve y2 + 4x - 6y + 13 = 0 is

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

VITEEEVITEEE 2018

Solution:

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