Find Equations for Differentials of 9y^2=x^3+3x^2 at (1,2/3), (-2,2/3), (-3,0)

[erik05]

If anyone could check this answer for me, it would be greatly appreciated.

Find the equations relating the differentials on the curve $$9y^2= x^3 +3x^2$$ at the points $$(1,\frac{2}{3}), (-2,\frac{2}{3}), (-3,0)$$

Here's what I got:

$$y'= \frac {x^2+2x}{6y}$$

$$m @ (1,\frac{2}{3})= \frac {3}{4}$$
equation: 9x-12y-1=0

$$m @ (-2,\frac{2}{3})=0$$
equation: y= 2/3

$$m @ (-3,0)$$
does not exist

Thanks in advance.

 

[xanthym]

If anyone could check this answer for me, it would be greatly appreciated.

Find the equations relating the differentials on the curve $$9y^2= x^3 +3x^2$$ at the points $$(1,\frac{2}{3}), (-2,\frac{2}{3}), (-3,0)$$

Here's what I got:

$$y'= \frac {x^2+2x}{6y}$$ <------ CORRECT

$$m @ (1,\frac{2}{3})= \frac {3}{4}$$
equation: 9x-12y-1=0 <----- CORRECT

$$m @ (-2,\frac{2}{3})=0$$
equation: y= 2/3 <----- CORRECT

$$m @ (-3,0)$$
does not exist <----- Should be the Vertical Line {x = -3}

Thanks in advance.

Everything appears correct except the last item.

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[erik05]

Thanks for the help.