Hyperbola
Intersection with Directrix
IIT-JEE 2010 Paper 1
Grade 11
Question:
The line $2x + y = 1$ is tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If this line passes through the point of intersection of the nearest directrix and the $x$-axis, then the eccentricity of the hyperbola is _____.
Step-by-Step Solution
Key Concept: Nearest directrix intersects $x$-axis at $(a/e, 0)$. It lies on $2x + y = 1$: $2a/e = 1 \Rightarrow a = e/2$. Tangency: $c^2 = a^2m^2 + b^2...$ wait, $2x + y = 1 \Rightarrow y = -2x + 1, m = -2$. Condition: $1 = 4a^2 - b^2 = 4a^2 - (c^2 - a^2) = 5a^2 - c^2$. With $a = e/2$ and $c = ae$: $1 = 5e^2/4 - e^4/4$. Let $u = e^2$: $u^2 - 5u + 4 = 0 \Rightarrow u = 4$ or $u = 1$. $e = 2$ (valid since $e > 1$).
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Correct Answer: 2