Differentiation, tangents and perpendiculars - A Level Question

[ahmad_03a]

Hi, I was just doing some basic differentiation revision and came across this question. I’m stuck on part C. I understand the method but when solving for the x coordinate of Q using the negative reciprocal of P as the gradient, I get two solutions (one negative and one positive). The solution (lined paper scan) says x is positive and I can’t seem to understand why... I might be missing a simple concept like always. I tried to visualise the graph of the tangent at point P on C but didn’t get anywhere. Attached is my working in iPad handwriting, the solution (on lined paper) and the question. Thanks in advance.

 

[Metume]

You get both positive and negative answers for 1/3(2+-root(6)) but you go with the positive version because the curve only allows for all x values greater than 0. 1/3(2-root(6)) is not greater than zero hence you go with the positive value. That is what the x>0 because the curve equation means. So yeah, simple concept overthought. Can relate though haha

 

[ahmad_03a]

You get both positive and negative answers for 1/3(2+-root(6)) but you go with the positive version because the curve only allows for all x values greater than 0. 1/3(2-root(6)) is not greater than zero hence you go with the positive value. That is what the x>0 because the curve equation means. So yeah, simple concept overthought. Can relate though hahaView attachment 27811

Thank you!!!