Question #309477

trace the curve (x^2+y^2)x=2x62+y^2

1
Expert's answer
2022-03-19T02:42:05-0400

1) Origin: Equation does not contain any constant term. Therefore, it passes through the origin.

The curve meets the coordinate axes only at the origin.

2) Symmetric about x-axis: The curve is symmetrical about x-axis, since only even powers of y occur.

3) Tangent at the origin: Equation of the tangent is obtained by equating to zero the lowest degree terms in the equation



2ay2xy2=x32ay^2-xy^2=x^3

Equation of the tangent:



2ay2=0y2=0,y=0 is a double point2ay^2=0\Rightarrow y^2=0, y=0\ is\ a\ double\ point

4) Cusp: As two tangents are coincident, therefore, origin is a cusp.

5) Asymptote parallel to y-axis: Equation of asymptote is obtained by equating the coefficient of highest degree of y to zero.



y2(2ax)=x32ax=0x=2ay^2(2a-x)=x^3\Rightarrow 2a-x=0 \Rightarrow x=2a

6) Region of absence of curve: y2y^2 becomes negative on putting x>2ax>2a or x<0,x<0, therefore, the curve does not exist for x<0x<0 and x>2a.x>2a.

(x^2+y^2)x=2x62+y^2

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