Hyperbola
Equation from Foci
Premium Question
Grade 11
Question:
The equation of the hyperbola whose foci are $(6, 4)$ and $(-4, 4)$ and eccentricity $2$ is:
(1) $\frac{(x - 1)^2}{25/4} - \frac{(y - 4)^2}{75/4} = 1$
(2) $\frac{(x - 1)^2}{25/4} - \frac{(y - 4)^2}{75/4} = -1$
(3) $\frac{(x - 1)^2}{75/4} - \frac{(y - 4)^2}{25/4} = 1$
(4) $(x - 1)^2 - \frac{(y - 4)^2}{3} = 1$
Step-by-Step Solution
Key Concept: Centre: midpoint $(1, 4)$. $c = 5$ (half distance $10/2$). $e = 2 \Rightarrow a = c/e = 5/2, a^2 = 25/4$. $b^2 = c^2 - a^2 = 25 - 25/4 = 75/4$. Equation: $(x - 1)^2/(25/4) - (y - 4)^2/(75/4) = 1$.
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Correct Answer: (1)