Find the focus of the parabola y = x² + x + 1.

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Q. Find the focus of the parabola y = x² + x + 1.

West Bengal WbchseWest Bengal HS First Year (WBCHSE Class XI) Annual Examination 2018Subjective· 2mImportance★★★★★est

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Completing the square converts $y=x^2+x+1$ to the standard parabola form $X^2=4pY$ , from which the focus follows.

Complete the square: $y=x^2+x+1=\left(x+\dfrac12\right)^2+\dfrac34$ , so

$\left(x+\dfrac12\right)^2=y-\dfrac34.$

Let $X=x+\dfrac12,\ Y=y-\dfrac34$ : this is $X^2=Y$ , i.e. $X^2=4\left(\dfrac14\right)Y$ , matching $X^2=4pY$ with $p=\dfrac14$ .

Vertex: $(X,Y)=(0,0)\implies (x,y)=\left(-\dfrac12,\dfrac34\right)$ .

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Q.Find the focus of the parabola y = x² + x + 1.