Question

The equation a(x⁴ + y⁴) – 4bxy (x² – y²) + 6c x²y² = 0

represents two pairs of lines at right angles. The two pairs will concide if-

• A. b² = a + 3c
• B. a² = b² – 3ac
• C. a² + b² = 3ac
• D. 2b² = a² + 3ac

Answer 1

Answer: (D)

2b² = a² + 3ac

Answer 2

The equation is homogeneous equation of fourth degree it must represent four straight lines passing through origin. The lines are perpendicular in pair. So

a(x⁴ + y⁴) – 4xy (x² – y²) + 6cx²y² = (ax² + pxy – ay²)

(x² + qxy – y²), p and q being constants.

On comparing similar power, we get

p + aq = –4b … (1)

and –2a + pq = 6c … (2)

Again, if two pairs coincide $\frac{p}{a}$= q Ž p = aq … (3)

From (1) and (3) q = –$\frac{2b}{a}$ and p = – 2b Ž from (2)

– 2a + $\frac{4b^{2}}{a}$ = 6c Ž 2b² = a² + 3ac.