Answer to Question #132035 in Calculus for Navya Sharma

Question

Find the angle between the curves y^2=ax and ay^2=x^3
(a>0
), at the points of
intersection other than the origin.

Answer 1

We shall find the angle between the curves y ²= ax and ay² = x ³

Consider the equations of the curves;

y ² = ax and ay² = x ³

Rewrite the equations

y ² = ax and y ² = x³/a,

ax = x³/a,

x ² = a ²,

x = a,

y = (ax) ^1/2 = x.

Since x = a; y = a,

the points of contacts = (a, a)

We differentiate

y ² = ax,

2y dy/dx= a,

dy/dx= a/(2y),  

M1 = 1/2,

ay² = x ³,

2ay  dy/dx= 3x ²,

dy/dx = 3/2

M2 = 3/2 

So;

tanᶱ = | (m1 – m2)/ (1 + m1*m2) |

      = | ( 1/2–3/2  )/ (1 + ½*3/2)

     = |-4/7|    = 4/7

ᶱ =tan^-1(4/7 )

ᶱ = 29.74◦