Prove that the eqation $xy + yz + zx = 0$ cuts the sphere $x^2 +y^2 + z^2 = a^2$ in two circle of equal area.

Question

Prove that the eqation $xy + yz + zx = 0$ cuts the sphere $x^2 +y^2 + z^2 = a^2$ in two circle of equal area.

Hint: on putting the value of $z= -\frac{xy}{x+y}$ in sphere. We get

$(x^2 +y^2)^2 + 2xy(x^2 + y^2) + x^2y^2 = a^2(x^2 + y^2)$

now the problem is to find radius of the two circle from the above equation ??

If any other methods to proof these , kindly send me.

Answer 1

HINT: $x^2+y^2+z^2+2(xy+yz+zx)=(x+y+z)^2$