Question 2
The two curves x3 β 3xy2 + 2 = 0 and 3x2y β y3 = 2
(A) touch each other (B) cut at right angle
(C) cut at an angle π/3 (D) cut at an angle π/4
Angles between two curves is same as angle between their tangents.
So, first we will find slope of their tangents.
Finding slope of tangent of first curve
π₯^3β3π₯π¦^2+2=0
Differentiating w.r.t x
γ3π₯γ^2β 3π¦^2 β 6xy ππ¦/ππ₯ = 0
γ3π₯γ^2β 3π¦^2 = 6xy ππ¦/ππ₯
ππ¦/ππ₯ = (3π₯^2 β3π¦^2)/6π₯π¦
π π/π π = (π^π βπ^π)/πππ
Let m1 = (π₯^2 βπ¦^2)/2π₯π¦
Finding slope of tangent of second curve
3π₯^2 π¦βπ¦^3β2=8
Differentiating w.r.t x
3π₯^2 ππ¦/ππ₯+6π₯π¦ β 3π¦^2 ππ¦/ππ₯ = 0
(3π₯^2β3π¦^2 ) ππ¦/ππ₯ = β6xy
ππ¦/ππ₯ = (β6π₯π¦)/(3π₯^2 β3π¦^2 )
π π/π π = (βπππ)/(π^π βπ^π )
Let, π_π= (β2π₯π¦)/(π₯^2 βπ¦^2 )
Finding Product of m1 & m2
m1 Γ m2 = (π₯^2 β π¦^2)/2π₯π¦ Γ ((β2π₯π¦)/(π₯^(2 )β π¦^2 ))
= β1
Since, product of the slopes is β1
β΄ Angle between tangents is 90Β°
Thus, curves cut each other at right angle.
So, the correct answer is (B)

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.