For the curve `y=f(x)` prove that (lenght of normal)^2/(lenght of tangent)^2 =length of sub-normal/length of sub-tangent

For a curve (length of normal)^2/(length of tangent)^2 is equal to

Consider curve y=f(x), then if PT = Length of tangent PN= Length of normal TM=Length of subtangent MN=Length of sub normal, then (a) T MdotM N=f^(prime)(x) (b) (P T)/(P N)=1/(f^(prime)(x)) (c) C TdotP N=con s t a n t (d) P TdotT M=con s t a n t

Prove that for the curve y=be^(x//a) , the subtangent is of constant length and the sub-normal varies as the square of the ordinate .

For the curve y=a1n(x^2-a^2) , show that the sum of length of tangent and sub-tangent at any point is proportional to product of coordinates of point of tangency.

For the curve y=a1n(x^2-a^2) , show that the sum of length of tangent and sub-tangent at any point is proportional to product of coordinates of point of tangency.

Find all the curves y=f(x) such that the length of tangent intercepted between the point of contact and the x-axis is unity.

Show that for the curve b y^2=(x+a)^3, the square of the sub-tangent varies as the sub-normal.

Find the length of sub-tangent to the curve y=e^(x//a)

In the curve x^(m+n)=a^(m-n)y^(2n) , prove that the m t h power of the sub-tangent varies as the n t h power of the sub-normal.

Let P be any point on the curve x^(2//3)+y^(2//3)=a^(2//3). Then the length of the segment of the tangent between the coordinate axes in of length