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Question

Prove that the equation y2+2Ax+2By+C=0 represents a parabola, whose axis is parallel to the axis of x, and find its vertex and the equation to its latus rectum.

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Solution

y2+2Ax+2By+C=0y2+2By=2AxCy2+2By+B2=2AxC+B2(y+B)2=2A(x+C2AB22A)

Comparing with the standard equation Y2=4aX

Axis of the parabola is Y=0

y+B=0y=B

Slope of the vertex =m=0

Hence it is parallel to x axis

Vertex of the parabola is X=0,Y=0

x+C2AB22A=0,y+B=0x=B2C2A,y=B(B2C2A,B)

Equation of the latusrectum is x=a

Here 4a=2Aa=A2

x+C2AB22A=A2x=B22AC2AA2x=B2CA22A


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