If the tangent at $P(m^2,m^3)$ on the curve $y^2=x^3$ is also the normal to same curve then what is the value of $9m^2$?

Question

If the tangent at $P(m^2,m^3)$ on the curve $y^2=x^3$ is also the normal to same curve then what is the value of $9m^2$?

My Approach: I considered another parametric point and the parameter is t. So the point is Q(t^2,t^3). Now I equated the slope of normal at Q with the slope of tangent at P and obtained the value of t in terms of m and the value is t=-4/9m and further obtained the answer as 9(m^2)=9

Now what my professor did was that he wrote the equation o tangent at P and satisfied the coordinates of Q in the equation and he obtained a quadratic equation in t. After solving the quadratic for the values of t in terms of m he got the value of t as t=-m/2 and got the answer 9(m^2)=8

So why are we getting different values of t? Whose approach is wrong? Or are both of us correct and the question will have 2 answers?

Please help

Answer 1

You have done a typo mistake. It should be $m^3$ instead of $m^2$. You can also another approach for this question here