# Identify the conic section given by displaystyle{y}^{2}+{2}{y}={4}{x}^{2}+{3} Find its frac{text{vertex}}{text{vertices}} text{and} frac{text{focus}}{text{foci}}

Identify the conic section given by ${y}^{2}+2y=4{x}^{2}+3$
Find its
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Step 1
We rewrite the equation as:
${y}^{2}+2y=4{x}^{2}+3$
${y}^{2}+2y-4{x}^{2}=3$
${y}^{2}+2y+1-4{x}^{2}=3+1$
${\left(y+1\right)}^{2}-4{x}^{2}=4$
$\frac{{\left(y+1\right)}^{2}}{4}-\frac{4{x}^{2}}{4}=1$
$\frac{{\left(y+1\right)}^{2}}{4}-\frac{{x}^{2}}{1}=1$
This is an equation of a hyperbola.
Step 2
Then we compare the equation with standard form.
$\frac{{\left(y+1\right)}^{2}}{4}-\frac{{x}^{2}}{1}=1$
$\frac{{\left(y-k\right)}^{2}}{{b}^{2}}-\frac{{\left(x-h\right)}^{2}}{{a}^{2}}=1$
$h=0,k=-1,$
${a}^{2}=1,{b}^{2}=4$
$a-1,b=2$