Relation between square of sub-tangent and sub-normal of $y^2 = (x+a)^3$

Question

If the relation between sub-tangent and sub-tangent at any point on the curve $y^2 = (x+a)^3$ is $p(SN)=q(ST)^2$ then find the value of $p:q$ where $SN$ is length of sub-normal and $ST$ is length of sub-tangent respectively.

Answer 1

Given that $SN$ is length of sub-normal and $ST$ is length of sub-tangent. Let us find slope which is nothing but $dy/dx$ at any point. Let the point be $(f,g)$ $y^2 = (x+a)^3$ $ 2 y dy/dx = 3(x+a)^2$ $dy/dx=[3(x+a)^2]/2y$ $dy/dx at (f,g) = 3(f+a)^2 /2g=m$ $SN=g.m= 3(f+a)^2 /2$ $ST=g/m= 2g^2 / [3(f+a)^2] = 2/3 (f+a)$ $(ST)^2 = 4/9 (f+a)^2 =4/9 . 2/3 (SN)$ $(ST)^2 = 8/27 (SN)$ $8(SN)=27(ST)^2$ $p$=8 and $q$=27 So $p:q$= 8$:$27