MHB Differential Equation: Tangents & Normals to y=x^2

[rozin]

Problem:Find the differential equation satisfied (i) by the equation of the family of tangents to y=x^2 and (ii) by the equation of the family of normals to y=x^2.

 

[HallsofIvy]

I presume you know that a tangent line to a graph has slope equal to the derivative of the function at the point of tangency. Surely you know that the derivative of y= x^2, at x= a, is y'= 2a. So what is the equation of the tangent line there? What differential equation does every such tangent line satisfy?

Do you know that a line "normal" to a graph is perpendicular to the tangent line at that point? And that the slope of a line, perpendicular to a line with slope "m", is -1/m?