First slide

Hyperbola in conic sections

Question

The two conics _{$\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1$} and _{$y^{2} = - \frac{b}{a} x$} intersect iff

Easy

A

_{$0 < a \leq \frac{1}{2}$}
B

_{$0 < b \leq \frac{1}{2}$}
C

_{$b^{2} > a^{2}$}
D

_{$b^{2} < a^{2}$}

Solution

The x co-ordinates of the points of intersection are given by _{$\frac{x^{2}}{a^{2}} + \frac{1}{a b} x + 1 = 0$}
And the roots are real iff _{$\frac{1}{a^{2} b^{2}} - \frac{4}{a^{2}} \geq 0 \Rightarrow \frac{1}{b^{2}} - 4 \geq 0$}

_{$\Rightarrow 0 < b^{2} \leq \frac{1}{4} \Rightarrow 0 < b \leq \frac{1}{2}$}

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