For the curve `y=f(x)` prove that (lenght n or mal)^2/(lenght or tanght)^2

Tangents are drawn from the origin to the curve y = sin x . Prove that their points of contact lie on the curve x^(2) y^(2) = (x^(2) - y^(2))

Tangents are drawn from the origin to curve y=sin x. Prove that points of contact lie on y^(2)=(x^(2))/(1+x^(2))

Let f(x) be a differentiable function and f(x)>0 for all x. Prove that the curves y=f(x) and y=sin kxf(x) touch each other at the points of intersection of the two curves.

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

If (-1,2) and and (2,4) are two points on the curve y=f(x) and if g(x) is the gradient of the curve at point (x,y) then the value of the integral int_(-1)^(2) g(x) dx is

Let us consider a curve, y=f(x) passing through the point (-2,2) and the slope of the tangent to the curve at any point (x,f(x)) is given by f(x)+xf'(x)=x^(2) . Then :

A curve y=f(x) passing through origin and (2,4). Through a variable point P(a,b) on the curve,lines are drawn parallel to coordinates axes.The ratio of area formed by the curve y=f(x),x=0,y=b to the area formed by the y=f(x),y=0,x=a is equal to 2:1. Equation of line touching both the curves y=f(x) and y^(2)=8x is