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For the pair of curves ax² + by² = 1 and a₁x² + b₁y² = 1 to intersect at right angles, which relation must hold?

1. (a-a₁)/(aa₁) = (b-b₁)/(bb₁)
2. (a+a₁)/(aa₁) = (b+b₁)/(bb₁)
3. (a-a₁)/(a+a₁) = (b-b₁)/(b+b₁)
4. None of these

Correct answer: (a-a₁)/(aa₁) = (b-b₁)/(bb₁)

Solution

At an intersection, slopes give (ax/by)(a1x/b1y)=-1, and subtracting the conics gives x^2/y^2 = -(b-b1)/(a-a1). Combining yields aa1(b-b1)=bb1(a-a1), i.e. (a-a1)/(aa1) = (b-b1)/(bb1).

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