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Consider the circle S in the xy-plane given by the equation x² + y² = 4. Chords E₁E₂ and F₁F₂ of S pass through the point P₀(1, 1) and are parallel to the x-axis and y-axis, respectively. Another chord G₁G₂ passes through P₀ and has a slope of −1. The tangents to S at E₁ and E₂ intersect at E₃, the tangents at F₁ and F₂ meet at F₃, and the tangents at G₁ and G₂ intersect at G₃. The points E₃, F₃, and G₃ lie on which curve?

1. x + y = 4
2. (x − 4)² + (y − 4)² = 16
3. (x − 4)(y − 4) = 4
4. xy = 4

Correct answer: x + y = 4

Solution

For x^2+y^2=4, the pole of chord y=1 is (0,4), of x=1 is (4,0), and of x+y=2 is (2,2). All three points satisfy x+y=4, so E3, F3, G3 lie on the line x+y=4, which is option (a), not (x-4)(y-4)=4.

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