China Mathematical Competition (Jiangxi)

Question

The curve represented by the equation x^22 - 3 + y^22 - 3 = 1 is ( ). (A) An ellipse with the foci on the x-axes (B) A hyperbola with the foci on the x-axes (C) An ellipse with the foci on the y-axes (D) A hyperbola with the foci on the y-axes

Accepted Answer

Since 2 + 3 > , so 0 < 2 - 2 < 3 - 2 < 2 and (2 - 2) > (3 - 2), i.e. 2 > 3. Since (2 - 3) - (2 - 3) = 222-32(2+32 + 4) (*) and -2 < 2-32 < 0, we get 2-32 < 0, 2 < 2+32 < 34, 34 < 2+32 + 4 < , (2+32 + 4) > 0, so the expression (*) is less than 0. That is 2 - 3 < 3 - 2, therefore the curve is an ellipse with foci on the y-axes. Answer: C.