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

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.

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.

Whole surface area of a solid hemisphere is equal to the curved surface area of a solid sphere. Find the ratio of lenghts of radius of hemisphere and sphere.

The equation of the ellips is (x^(2))/(36)+(y^(2))/(16)=k. The differential equation of the ellips whose length of major and minor axes half the lenghts of the given ellipse respectively, is