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

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

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 the curve y = sin x . Prove that their points of contact lie on the curve x^2y^2=(x^2-y^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))

Sketch the curve y =f(x)= x^(2) - 5x + 4

Sketch the curve y = f(x) = x^(2) -5x + 6

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.