Question

Find the equation of the quartic curve which has x = 0, y = 0, y = x, and y = –x four asymptotes which pass through (a, b) and which cuts the curve in eight points that lie on the circle x2 + y2 = a2.

Answer 1

The ***** equation of the asymptotes, i.e. Fn = 0 is

xy(y – x)(y + x) = 0 or xy(x2 – y2) = 0 ....(1)

The given equation of common points of intersection is x2 + y2 – a2 = 0 ....(2)

Here the equation of the quartic whose asymptotes are given by (1) and whose intersection with the asymptotes lie on (2), is given by

Fn + Fn – 2 = 0, i.e. xy(x2 – y2 ) + λ(x2 + y2 – a2) = 0 ....(3)

whence λ is a constant.

Now this curve pass through the point (a, b) means

ab(a2 – b2 ) + λ(a2 + b2 – a2 ) = 0 or

λ = a(a2 - b2)/b ...(4)

Whence with above value of λ, the equation of the quartic becomes

b xy(x2 – y2 ) + a(a2 – b2)(x2 + y2 – a2 ) = 0.