Hyperbola
Equation of Hyperbola
Premium Question
Grade 11
Question:
Let the eccentricity of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be the reciprocal of the eccentricity of the ellipse $x^2 + 4y^2 = 4$. If the hyperbola passes through a focus of the ellipse, then it is:
(1) $\frac{x^2}{3} - y^2 = 1$
(2) $\frac{x^2}{2} - \frac{y^2}{2} = 1$
(3) $3x^2 - y^2 = 3$
(4) $x^2 - 3y^2 = 3$
Step-by-Step Solution
Key Concept: Ellipse $x^2/4 + y^2/1 = 1$: $e_E = \sqrt{3}/2$, foci $(\pm \sqrt{3}, 0)$. Hyperbola: $e_H = 2/\sqrt{3}$, so $1 + b^2/a^2 = 4/3 \Rightarrow b^2 = a^2/3$. Passing through $(\sqrt{3}, 0)$: $3/a^2 = 1 \Rightarrow a^2 = 3$, $b^2 = 1$. Equation: $x^2/3 - y^2 = 1$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)