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| Conic
Sections |
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Hyperbola
and Line
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 Hyperbola and line relationships
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Condition for a line to be the tangent to the hyperbola -
tangency condition |
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The equation of the tangent at the point on the hyperbola
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Polar and pole of the hyperbola
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Construction of the tangent at the point on the hyperbola
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| Hyperbola and line relationships
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| Let examine relationships between a hyperbola and a line passing through the center of the hyperbola, i.e., the
origin. A line y =
mx intersects the hyperbola at two
points if
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the slope |m|
<
b/a but
if |m|
> b/a
then,
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the line y =
mx does not intersect the hyperbola at all.
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| The diameters of a hyperbola are straight lines
passing through its center.
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The asymptotes divide these two pencils of diameters
into, one which intersects the curve at two points, and
the
other which do not intersect.
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A diameter of a conic section is a line which passes
through the midpoints of parallel chords.
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Conjugate diameters of the hyperbola (or the ellipse)
are two diameters such that each bisects all chords
drawn parallel to the other.
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| As the equation of a hyperbola can be obtained from the equation of an ellipse by changing the sign of
b2
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that is, a2(1
-
e2)
= -a2(e2
-
1) =
-b2, |
| this way, we can use other formulas relating to the ellipse to obtain corresponding
formulas for the hyperbola. |
| Therefore, when we examine conditions which determine position of a line in relation to a hyperbola that is, |
| when
solve the system of
equations,
y =
mx + c |
| b2x2
-
a2y2
= a2b2
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then if, a2m2
-
b2
> c2
the line intersects the hyperbola at two points,
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a2m2
-
b2 = c2
the line is the tangent of the hyperbola,
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a2m2
-
b2
< c2
the line and the hyperbola do not intersect. |
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| Condition for a line to be the tangent to the hyperbola -
tangency condition
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| A line is the tangent to the hyperbola if
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a2m2
-
b2 = c2. |
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| Regarding
the asymptotes, to which c
= 0, this condition gives
|m|
= b/a
and that is way we can say that the hyperbola touches the asymptotes at infinity. |
| From the tangency condition it also follows that the slopes of the tangents
will satisfy the condition, |
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| That is, the tangents to the hyperbola can only be parallel to a line belonging to the pencil of lines that do not
intersects the hyperbola. |
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| The equation of the tangent at the point on the hyperbola
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As we already mentioned, the points of contact of a line and the hyperbola can be obtained from the
corresponding formula for the ellipse by changing b2
with -b2
thus |
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the
tangency point or the point of contact. |
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| So, the intercept and slope of the
tangent |
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| or |
b2x1x
-
a2y1y
= a2b2 |
the equation of the tangent at a point
P1(x1,
y1)
on the hyperbola.
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| Polar and pole of the hyperbola
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If from a point
A(x0, y0), exterior to the hyperbola, drawn are tangents, then the secant line passing through
the contact points, is the polar of the point A.
The point A
is called the pole of the polar. |
| The equation of the
polar is derived the same way as for the ellipse, |
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b2x0x
-
a2y0y
= a2b2 |
the equation of the
polar of the point A(x0, y0). |
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| Construction of the tangent at the point on the hyperbola
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| The tangent at the point
P1(x1,
y1)
on the hyperbola is the bisector of the angle F1P1F2
subtended by focal |
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radii,
r1
and
r2 at
P1 .
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The proof shown for the ellipse applied to
the hyperbola gives,
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| or |
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See the title
'
The angle between the focal radii at a point of the
ellipse'.
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| College
algebra contents E |
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