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F1 and F2 are the foci of the hyperbola x²/16 - y²/9 = 1, while F3 and F4 are the foci of its conjugate hyperbola. Let e_H and e_C denote the eccentricities of the hyperbola and its conjugate, respectively. Which statement is true?
- Their equations of the asymptotes are different.
- e_H > e_C
- Area of the quadrilateral formed by their foci is 50 sq. units.
- Their auxiliary circles will have the same equation.
Correct answer: Area of the quadrilateral formed by their foci is 50 sq. units.
Solution
Both hyperbolas share asymptotes (so option A false). c_H = sqrt(16+9)=5 on x-axis; c_C = sqrt(9+16)=5 on y-axis; the four foci (±5,0),(0,±5) form a square of diagonal 10, area = (1/2)*10*10 = 50 sq. units.
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