| 1. | This curve has a tangent line at the origin that is vertical.
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| 2. | Drawing a set of such tangent lines reveals the envelope of the hyperbola.
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| 3. | But only a tangent line is perpendicular to the radial line.
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| 4. | Geometrically, the limit of the secant lines is the tangent line.
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| 5. | The next step is to find the intersection points of the tangent lines.
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| 6. | Any such jump or point discontinuity will have no tangent line.
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| 7. | A point on just one tangent line is on the conic.
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| 8. | Tangent lines to a hyperbola have another remarkable geometrical property.
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| 9. | Conversely, the perpendicular to a radius through the same endpoint is a tangent line.
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| 10. | For this reason, this process is also called the "'tangent line approximation " '.
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