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)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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