Answer to Question #309477 in Algebra for ANSARI

Question

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

Answer 1

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

$$2ay^2-xy^2=x^3$$

Equation of the tangent:

$$2ay^2=0\Rightarrow y^2=0, y=0\ is\ a\ double\ point$$

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.

$$y^2(2a-x)=x^3\Rightarrow 2a-x=0 \Rightarrow x=2a$$

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

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