When the origin is shifted to $(h, k)$ by translation of axes, the transformed equation of $x^2+2 x+2 y-7=0$…

When the origin is shifted to $(h, k)$ by translation of axes, the transformed equation of $x^2+2 x+2 y-7=0$ does not contain $x$ term and constant term. Then $(2 h+k)=$
  1. $\frac{7}{2}$
  2. $\frac{1}{2}$
  3. 2
  4. 0

Solution

$\begin{aligned} & \text { } \mathrm{X}+h=x, \mathrm{Y}+k=y \\ & (\mathrm{X}+h)^2+2(\mathrm{X}+h)+2(\mathrm{Y}+k)-7=0 \\ & \mathrm{X}^2+(2 h+2) \mathrm{X}+2 \mathrm{Y}+h^2+2 h+2 k-7=0 \end{aligned}$
To remove X - term, $2 h+2=0 \Rightarrow h=-1$ To remove constant term, $h^2+2 h+2 k-7=0$ $\Rightarrow 1-2+2 k-7=0 \Rightarrow k=4 \Rightarrow 2 h+k=-2+4=2$

Asked in: AP EAMCET 2024 (21 May Shift 1)

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