Question

19. Find the asymptotes, parallel to the axes for the curve y4 + x2y2 + 2xy? – 4x2 - y +1=0
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Answer 1

Step-by-step explanation:

Rewrite the given equation as

x2y – xy2 – xy + y2 + x – y = 0

as yx2 + (1 – x)y2 – xy + x – y = 0

Clearly asymptotes parallel to X-axis and Y-axis are y = 0 and x = 1 respectively. For oblique asymptotes:

φ3(m) = m – m2 and φ2(m) = –m + m2 implying φ'3(m) = 1 – 2m

so that c =

Therefore, equation of asymptotes y = mx + c becomes y = 0 , y = x. Now the joint equation of the asymptotes,

y(x – 1)(y – x) = 0 or x2y – xy2 – xy + y2 = 0

Clearly given equation of the curve, i.e. (x2y – xy2 – xy + y2 ) + (x – y) = 0 is expressible like Fn + Fn – 2 = 0 wherein we have obtained, Fn = 0, i.e. x2y – xy2 – xy + y2 = 0 as the joint equation of the asymptotes.

and c = - φ2(m)/φ3(m) = - ( - 2am2)/ 2m = am = ± a

Therefore the two oblique asymptotes are y = x + a and y = –x – a.