First slide

Hyperbola in conic sections

Question

If a variable line has its intercepts on the coordinate axes e and e' , where $\frac{e}{2}$ and $\frac{e'}{2}$ are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle $x^{2} + y^{2} = r^{2}$ , where r =

Moderate

A

1
B

2
C

3
D

cannot be decided

Solution

Since $\frac{e}{2}$ and e $\frac{e'}{2}$ are the eccentricities of a hyperbola and
its conjugate hyperbola, we have
$\begin{aligned} \frac{4}{e^{2}} + \frac{4}{e^{' 2}} = 1 \\ or 4 = \frac{e^{2} e^{' 2}}{e^{2} + e^{' 2}} \end{aligned}$
The line passing through the points (e, 0) and (0, e') is
$e^{'} x + ey - {ee}^{'} = 0$
It is tangent to the circle $x^{2} + y^{2} = r^{2}$ Hence,
$\begin{array}{l} \frac{{ee}^{'}}{\sqrt{e^{2} + e^{' 2}}} = r \\ or r^{2} = \frac{e^{2} e^{' 2}}{e^{2} + e^{' 2}} = 4 \\ or r = 2 \end{array}$

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